AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Volume 6, number 1 December 2021

http://reportsascm.iec.cat ISSN electronic edition: 2385 - 4227

Editorial Team Editor-in-chief Xavier Bardina, Universitat Autònoma de Barcelona (stochastic analysis, probability)

Associate Editors Marta Casanellas, Universitat Politècnica de Catalunya (algebraic geometry, phylogenetics) Pedro Delicado, Universitat Politècnica de Catalunya (statistics and operations research) Alex Haro, Universitat de Barcelona (dynamical systems) David Marı́n, Universitat Autònoma de Barcelona (complex and differential geometry, foliations) Xavier Massaneda, Universitat de Barcelona (complex analysis) Eulàlia Nualart, Universitat Pompeu Fabra (probability) Joaquim Ortega-Cerdà, Universitat de Barcelona (analysis) Francesc Perera, Universitat Autònoma de Barcelona (non commutative algebra, operator algebras) Julian Pfeifle, Universitat Politècnica de Catalunya (discrete geometry, combinatorics, optimization) Albert Ruiz, Universitat Autònoma de Barcelona (topology) Gil Solanes, Universitat Autònoma de Barcelona (differential geometry) Enric Ventura, Universitat Politècnica de Catalunya (algebra, group theory)

Focus and Scope Reports@SCM is a non-profit electronic research journal on Mathematics published by the Societat Catalana de Matemàtiques (SCM) which originated from the desire of helping students and young researchers in their first steps into the world of research publication. Reports@SCM publishes short papers (maximum 10 pages) in all areas of pure mathematics, applied mathematics, and mathematical statistics, including also mathematical physics, theoretical computer science, and any application to science or technology where mathematics plays a central role. To be considered for publication in Reports@SCM an article must be written in English (with an abstract in Catalan), be mathematically correct, and contain some original interesting contribution. All submissions will follow a peer review process before being accepted for publication. Research announcements containing preliminary results of a larger project are also welcome. In this case, authors are free to publish in the future any extended version of the paper elsewhere, with the only condition of making an appropriate citation to Reports@SCM. We especially welcome contributions from researchers at the initial period of their academic careers, such as Master or PhD students. We wish to give special attention to the authors during the whole editorial process. We shall take special care of maintaining a reasonably short average time between the reception of a paper and its acceptance, and between its acceptance and its publication. All manuscripts submitted will be peer reviewed by at least one reviewer. Final decisions on the acceptance of manuscripts are taken by the editorial board, based on the reviewer’s opinion.

This work is subject to a Recognition - Non Commercial - Without derivative works Creative Commons 3.0 Spain license, unless the text, pictures or other illustrations indicate the contrary. License’s full text can be read at http://creativecommons.org/licenses/by-nc-nd/3.0/es/deed.ca. Readers can reproduce, distribute and communicate the work as long as its authorship and publishing institution are recognized and also if this does not entail commercial use or derivative work. c The authors of the articles Edited by Societat Catalana de Matemàtiques, Institut d’Estudis Catalans (IEC) Carrer del Carme 47, 08001 Barcelona. http://scm.iec.cat Telèfon: (+34) 93 324 85 83 scm@iec.cat Fax: (+34) 93 270 11 80 Style revision by Enric Ventura. Institut d’Estudis Catalans http://www.iec.cat informacio@iec.cat http://reportsascm.iec.cat ISSN electronic edition: 2385-4227

Reports@SCM Volume 6, number 1, 2021

Table of Contents

Factor analysis: Existence of solution to factor models Adrià Prior Rovira

1

Hirzebruch signature theorem and exotic smooth structures on the 7-sphere Guifré Sánchez Serra

11

Random walks on supersingular isogeny graphs Enric Florit Zacarı́as

23

Self-similar profiles in Analysis of Fluids. A 1D model and the compressible Euler equations Gonzalo Cao-Labora

35

Homotopical realizations of infinity groupoids Jan McGarry Furriol

49

AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Factor analysis: Existence of solution to factor models ∗ Adrià

Prior Rovira

Universitat Autònoma de Barcelona adriaprior2@gmail.com ∗Corresponding author

Resum (CAT) Un dels resultats principals de l’anàlisi factorial afirma que si el model factorial se satisfà per a un vector aleatori X , aleshores la matriu de covariància del vector admet una descomposició en termes de les matrius que caracteritzen el model, i que el recı́proc també és cert sota condicions força generals. La implicació directa es troba demostrada en moltes referències, però la prova del recı́proc sembla difı́cil de trobar en els textos disponibles. La raó d’aquest article és compartir una demostració del recı́proc concebuda per l’autor, primerament pel cas del model factorial ortogonal, àmpliament usat en anàlisi factorial exploratòria i, en segon lloc, per un model factorial que generalitza l’ortogonal i està pensat per a ser utilitzat en anàlisi factorial confirmatòria.

Abstract (ENG) One of the main results in factor analysis states that if the factor model holds for a random vector X , then the covariance matrix of the vector admits a decomposition in terms of the matrices that characterize the model, and that the converse is also true under quite general conditions. The direct implication can be found proved in many references but the proof of the converse seems difficult to find in the available texts. The reason of this article is to share an original proof of the converse, first for the case of the orthogonal factor model, widely used in exploratory factor analysis, and secondly for a factor model that generalizes the orthogonal one, and which is meant to be used in confirmatory factor analysis.

Keywords: factor analysis, exploratory factor analysis, confirmatory factor analysis, factor model, fundamental theorem of factor analysis. MSC (2010): Primary 55M25, 57P10. Secondary 55P15, 57R19, 57N15. Received: March 8, 2021. Accepted: May 17, 2021. http://reportsascm.iec.cat

Acknowledgement I would like to thank the anonymous referees for carefully reading the manuscript and for their useful corrections and suggestions.

Reports@SCM 6 (2021), 1–9; DOI:10.2436/20.2002.02.23.

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Existence of solution to factor models

1. Introduction Factor analysis is a statistical theory that allows, under certain conditions, to express each variable of an observable random vector as a linear combination of a few new variables called factors, through a stochastic model. The so called factor model is adjusted to the observable vector, which we shall call initial vector, using a set of observations of it. This technique is used in different fields such as psychology, sociology, economy or political sciences, and also in the physical sciences and biosciences. The factors may explain the initial variables in a simplified way, and they should be of theoretical interest in the specific research setting. In fact, factor analysis is usually applied to investigate concepts that can’t be measured directly, like intelligence, social status or sustainable progress, by collapsing a large number of variables related to those concepts into a reduced group of latent factors. There exists a vast literature in factor analysis that treats the applied aspects of the technique, including a variety of examples; see Mardia et al. ([5, pp. 255–280]) for a consistent introduction. One of the main results in this frame says that if the factor model holds for the initial random vector, then its covariance matrix can be decomposed in a particular way, and that the converse is also true under quite general conditions. The direct implication is proven in many references, but the proof of the converse cannot be easily found in the available texts, despite it is stated in several ones. The author proved this result after unsuccessfully seeking a complete detailed proof, following a hint given in Mardia et al. [5]. The reason of this paper is to share this proof, so that it can be accessible to anybody interested in factor analysis from the mathematical point of view. First, we will prove the converse for the case of the orthogonal factor model, which is commonly used in exploratory factor analysis. Secondly, we will extend the result to a factorial model in which correlations between factors will be allowed, being this way adequate to use in confirmatory factor analysis, we will call this second model generalized factor model. Exploratory factor analysis is probably the most known version of the technique, and it is used to find a factor model that fits the initial vector, whereas the confirmatory variant is usually performed after an exploratory analysis, with the aim of fitting a specific factor model such that some of the parameters values are predetermined in advance by the researchers. We will name Theorem of existence of solution to the orthogonal factor model and Theorem of existence of solution to the generalized factor model the two results, which are precisely stated and proved in Sections 2 and 3, respectively. Both theorems are important in practice because they ensure that the factor model is valid when the sample covariance matrix fits well to a given pattern, which describes the relations between the initial variables and the latent unobserved factors, as well as the relations between the factors.

2. Existence of solution to the orthogonal factor model We begin with the orthogonal case, by defining what is a solution to the orthogonal factor model for a random vector X .

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Definition 2.1. Let X t = (X1 , ... , Xp ) be a p × 1 random vector with E[X ] = 0p×1 and E[X 2 ] < ∞1 . We say that the orthogonal factor model holds for X if there exist two random vectors f t = (f1 , ... , fm ) with m < p and u t = (u1 , ... , up ) and a matrix Q = (qij )ij ∈ Mp×m (R), such that X1 = q11 f1 + q12 f2 + · · · + q1m fm + u1 X2 = q21 f1 + q22 f2 + · · · + q2m fm + u2 (1) .. . Xp = qp1 f1 + qp2 f2 + · · · + qpm fm + up and satisfying the following conditions: (i) E[f ] = 0m×1 , Cov(f ) = Im , with Im the identity matrix on Rm . (ii) E[u] = 0p×1 , Cov(u) = Ψ, with Ψ a diagonal matrix in Mp (R). (iii) Cov(f , u) = 0m×p , where Cov(f , u) denotes the cross-covariance matrix between f and u. In this case we say that the triplet (Q, f , u) is a solution to the orthogonal factor model for X , (f1 , ... , fm ) are called the common factors of the model, (u1 , ... , up ) are called the specific factors, the matrix Q is called the loadings matrix and the elements on the diagonal of Ψ are named specific variances. The model equations system (1) can be written as X = Qf + u. In practice, X is the observed random vector for which we want to fit the model, the assumption E[X ] = 0p×1 is not restrictive since data can be centered to get the model and translated to the original center at the end, if necessary. We demand m < p because one of the objectives of factor analysis is explaining the initial variables in a simplified way with a few common factors. The specific factors can be understood as the stochastic error terms in regression. Condition (i) asks the common factors to be uncorrelated and have unit variance. Is in this sense that we call the model orthogonal, considering the covariance as a scalar product. Conditions (ii) and (iii) ask the specific factors to be uncorrelated one to each other and uncorrelated to the common factors. These last two assumptions seem natural, in the sense that the common factors capture and explain a part of the variability of each initial variable, letting the remaining amount to the specific factors. To clarify notation, we will use ΣX as well as Cov(X ) to denote the covariance matrix of a random vector X , depending on the situation, that is, ΣX = Cov(X ). The next proposition is sometimes called The fundamental theorem of factor analysis and it shows us a necessary condition for the model to have solution in the sense of Definition 2.1. Similar proofs as the given below can be found in the literature, for example in Härdle and Simar ([2, p. 310]). Proposition 2.2. Let X be a random vector with E[X ] = 0p×1 and E[X 2 ] < ∞. If (Q, f , u) is a solution to the orthogonal factor model for X , with Cov(u) = Ψ, then ΣX = QQ t + Ψ.

(2)

Proof. Using basic properties of the covariance matrix and Definition 2.1, it is clear that ΣX = Cov(X ) = Cov(Qf + u) = Cov(Qf ) + Cov(Qf , u) + Cov(u, Qf ) + Cov(u) = QQ t + Ψ. 1

Let X t = (X1 , ... , Xp ) be a p × 1 random vector. We say that E[X 2 ] < ∞ if E[Xj2 ] < ∞ for all j ∈ {1, ... , p}. We demand E[X ] < ∞ to ensure existence of the covariances between the variables in X . 2

Reports@SCM 6 (2021), 1–9; DOI:10.2436/20.2002.02.23.

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Existence of solution to factor models

The theorem of existence of solution we are interested in is the converse of the above proposition, and it states that condition (2) is sufficient provided that Ψ is positive definite. The proof of this result seems difficult to find in the existing literature. An original proof is given below, which uses some results on block matrices and basic knowledge on linear algebra and probability. Theorem 2.3 (Existence of solution to the orthogonal factor model). Let X be a p × 1 random vector with E[X ] = 0p×1 and E[X 2 ] < ∞. Assume that there exist two matrices Q ∈ Mp×m (R), with m < p, and Ψ ∈ Mp (R) diagonal and positive definite, such that ΣX = QQ t + Ψ. Then, there exist two random vectors f t = (f1 , ... , fm ) and u t = (u1 , ... , up ) satisfying the orthogonal factor model for X , with loadings matrix Q and Cov(u) = Ψ. Proof. Following the hint given in Mardia et al. ([5, p. 276]), we will show first that there exists a multivariate normal random vector Y t = (Y1 , ... , Ym ) with Y ∼ Nm (0m×1 , Im + Q t Ψ−1 Q), and then we will show that the pair of random vectors defined by −1 u Ip Q X := (3) f −Q t Ψ−1 Im Y | {z } A−1

are a solution to the orthogonal factor model. Take W := Im +Q t Ψ−1 Q, which is well defined since Ψ = diag(ψ1 , ... , ψp ) with ψi > 0, ∀ i ∈ {1, ... , p}. First of all, let’s see that W is symmetric and positive definite, and therefore we can consider a multivariate normal vector Y with covariance matrix W . Indeed, Q t Ψ−1 Q and Im are symmetric and hence W is. Let v ∈ Mm×1 (R) be any vector and take y := Qv . We have v t Q t Ψ−1 Qv = y t Ψ−1 y ≥ 0. Therefore v t Wv = v t v + v t Q t Ψ−1 Qv > 0 for any non null v ∈ Mm×1 (R), and W is positive definite. Now, let’s see that the matrix A in (3) is invertible. A is a square matrix and Im is invertible, hence Schur’s determinant formula (Schur, [6]) applies to obtain Ip Q det(A) = det = det(Im ) det(A/Im ) = det(Ip + QQ t Ψ−1 ) = det(Ψ + QQ t ) det(Ψ−1 ), −Q t Ψ−1 Im where A/Im = Ip + QQ t Ψ−1 is the Schur complement of Im in A. Then, A is invertible if and only if Ψ + QQ t = ΣX is. As QQ t is positive semidefinite and Ψ is positive definite, Ψ + QQ t is positive definite, so A is invertible. After this technical details, we are ready to prove that the factors in (3) give a solution to the model. Since A is invertible, we have: −1 u Ip Q X X Ip Q u = ⇐⇒ = . f −Q t Ψ−1 Im Y Y −Q t Ψ−1 Im f | {z } A

Clearly, X = Qf + u, we must show that f and u satisfy conditions (i), (ii) and (iii) of Definition 2.1. To see that (ii) holds observe that u X −1 X −1 E = EA =A E = 0(p+m)×1 f Y Y

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thus E[u] = 0p×1 and E[f ] = 0m×1 . Now, take M := Ip + QQ t Ψ−1 = A/Im . Since A and Im are invertible, M is too by the Schur’s determinant formula and we can apply a Banachiewicz inversion formula (Banachiewicz, [1]) to obtain the inverse −1

A

=

Ip Q t −1 −Q Ψ Im

−1

M −1 −M −1 Q . Q t Ψ−1 M −1 Im − Q t Ψ−1 M −1 Q

=

Thus, u M −1 −M −1 Q X −1 X =A = f Y Q t Ψ−1 M −1 Im − Q t Ψ−1 M −1 Q Y and therefore, u = M −1 (X − QY ).

(4)

Now, Cov(u) = Cov(M −1 (X − QY )) = M −1 Cov(X − QY )(M −1 )t , and developing the covariance: Cov(X − QY ) = Cov(X ) + Cov(X , −QY ) + Cov(−QY , X ) + Cov(−QY ) = ΣX + QΣY Q t = (Ψ + QQ t ) + Q(Im + Q t Ψ−1 Q)Q t = (Ip + QQ t Ψ−1 )(Ψ + QQ t ) = (Ip + QQ t Ψ−1 )Ψ(Ip + Ψ−1 QQ t ) = MΨM t . Where we have used ΣX = QQ t + Ψ by hypothesis, and Cov(X , Y ) = 0p×m since Y is taken independently of X , hence Cov(u) = M −1 MΨM t (M t )−1 = Ψ and (ii) is proven. Using similar arguments we will prove that Cov(f ) = Im . Provided that A and Ip are invertible we now use another Banachiewicz inversion formula to get the following expression: A−1 =

Ip Q −Q t Ψ−1 Im

−1 =

Ip − QW −1 Q t Ψ−1 −QW −1 , W −1 Q t Ψ−1 W −1

where W = Im + Q t Ψ−1 Q = A/Ip is the Schur complement of Ip in A. This way, u X Ip − QW −1 Q t Ψ−1 −QW −1 −1 X =A = Y f Y W −1 Q t Ψ−1 W −1 and we obtain f = W −1 (Q t Ψ−1 X + Y ).

(5)

Then, Cov(f ) = W −1 Cov(Q t Ψ−1 X + Y )(W −1 )t and using again that Cov(X , Y ) = 0p×m , it follows that Cov(Q t Ψ−1 X + Y ) = Q t Ψ−1 ΣX Ψ−1 Q + ΣY = Q t Ψ−1 (QQ t + Ψ)Ψ−1 Q + W = (Q t Ψ−1 Q + Im )(Q t Ψ−1 Q) + W = W (Q t Ψ−1 Q) + W = W (Q t Ψ−1 Q + Im ) = WW .

Reports@SCM 6 (2021), 1–9; DOI:10.2436/20.2002.02.23.

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Existence of solution to factor models

Therefore, using that W is symmetric, we have Cov(f ) = W −1 WW (W −1 )t = W −1 WWW −1 = Im and (i) holds. Finally, let us see that (iii) holds, that is, Cov(u, f ) = 0p×m . Using the expressions (4) and (5) we have: Cov(u, f ) = Cov(M −1 (X − QY ), W −1 (Q t Ψ−1 X + Y )) = M −1 Cov(X − QY , Q t Ψ−1 X + Y )(W −1 )t = M −1 [Cov(X , X )Ψ−1 Q − Q Cov(Y , Y )]W −1 = M −1 [ΣX Ψ−1 Q − QΣY ]W −1 , but the term in square brackets is null, that is, ΣX Ψ−1 Q − QΣY = (QQ t + Ψ)Ψ−1 Q − QW = Q(Q t Ψ−1 Q + Im ) − QW = QW − QW = 0p×m . So Cov(u, f ) = 0p×m , and the proof is complete. Under the hypotheses of Theorem 2.3 the existence of a solution holds, but the solution is not unique, in fact, every orthogonal matrix provides another solution. Next proposition states this result (see Mardia et al. for a proof, [5, pp. 257–258]). Proposition 2.4. Let X be a random vector with E [X ] = 0p×1 and E [X 2 ] < ∞, let m < p and let G ∈ Mm (R) be an orthogonal matrix, that is, G t G = GG t = Im . If (Q, f , u) is a solution to the orthogonal m-factor model for X , then (QG , G t f , u) is a solution too. In particular, the result holds when G ∈ Mm (R) is orthogonal and det(G ) = 1, that is, when G is a rotation matrix in Rm . Theorem 2.3 indicates how to proceed to adjust the model to a given observed vector X t = (X1 , ... , Xp ). e ∈ Mn×p (R), where each row of X e is an observation of X , and we In practice we have a data matrix X b ∈ Mp×m (R) and estimate ΣX by the sample covariance matrix S, then, our objective is to find matrices Q b ∈ Mp (R), with Ψ b diagonal and positive definite, such that the equality Ψ bQ bt + Ψ b S =Q b and Ψ, b they can be taken as loadings and specific holds, at least approximately. If we find such matrices Q variances estimates and so, they give rise to an estimated solution to the orthogonal factor model. b and Ψ b are found using numerical methods currently implemented in statistical software Estimates Q environments like R, being the Maximum Likelihood Estimation (MLE) and the least squares methods two popular examples (Jöreskog, [3] and [4]). In exploratory factor analysis, the solution estimated by these methods may not be useful enough for the researchers, in the sense that the factors may load on too many variables and it could be difficult to interpret them. In this case, there exist methods (varimax, orthomax and others) that aim to provide a rotation matrix G such that the rotated factors, given by Proposition 2.4, may be more relevant for the ongoing investigation. From this point of view, the non uniqueness of solution is not a drawback.

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3. The generalized factor model: existence of solution The generalized factor model will allow correlations between the common factors, which is a less restrictive and so more realistic assumption in many settings. We define: Definition 3.1. Let X t = (X1 , ... , Xp ) be a p × 1 random vector with E[X ] = 0p×1 and E[X 2 ] < ∞. We say that the generalized factor model holds for X if there exist two random vectors f t = (f1 , ... , fm ) with m < p and u t = (u1 , ... , up ) and a matrix Q = (qij )ij ∈ Mp×m (R), such that X1 = q11 f1 + q12 f2 + · · · + q1m fm + u1 X2 = q21 f1 + q22 f2 + · · · + q2m fm + u2 .. . Xp = qp1 f1 + qp2 f2 + · · · + qpm fm + up and satisfying the following conditions: (i) E[f ] = 0m×1 , Cov(f ) = Θ, with Θ ∈ Mm (R) symmetric and positive semidefinite. (ii) E[u] = 0p×1 , Cov(u) = Ψ, with Ψ a diagonal matrix in Mp (R). (iii) Cov(f , u) = 0m×p . In this case we say that the triplet (Q, f , u) is a solution to the generalized factor model for X . The common factors in f are also called “latent” or “hidden” factors for X . Proposition 3.2. Let X be a random vector with E [X ] = 0p×1 and E[X 2 ] < ∞. If (Q, f , u) is a solution to the generalized factor model for X , with Cov(f ) = Θ and Cov(u) = Ψ, then ΣX = QΘQ t + Ψ.

(6)

Proof. ΣX = Cov(X ) = Cov(Qf + u) = Q Cov(f )Q t + Q Cov(f , u) + Cov(u, f )Q t + Cov(u) = QΘQ t + Ψ.

Therefore, the necessary condition for the model to have solution is now ΣX = QΘQ t + Ψ, with Q ∈ Mp×m (R), m < p, and Θ and Ψ covariance matrices, with the second being diagonal. This condition is also sufficient as in the orthogonal case, if we ask Ψ to be positive definite. The result is given by the next theorem, which is a corollary of Theorem 2.3. Theorem 3.3 (Existence of solution to the generalized factor model). Let X be a p × 1 random vector with E [X ] = 0p×1 and E[X 2 ] < ∞. Assume that there exist three matrices Q ∈ Mp×m (R), with m < p, Ψ ∈ Mp (R), with Ψ diagonal and positive definite, and Θ ∈ Mm (R), with Θ symmetric and positive semidefinite, such that ΣX = QΘQ t + Ψ. Then, there exist two random vectors f t = (f1 , ... , fm ) and u t = (u1 , ... , up ) satisfying the generalized factor model for X , with loadings matrix Q, with Cov(f ) = Θ and Cov(u) = Ψ.

Reports@SCM 6 (2021), 1–9; DOI:10.2436/20.2002.02.23.

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Existence of solution to factor models

Proof. Assume ΣX = QΘQ t + Ψ, with Q ∈ Mp×m (R), Θ ∈ Mm (R) and Ψ ∈ Mp (R), with the stated conditions. Since Θ is symmetric, we can consider its spectral decomposition Θ = V ΛV t . Then, Λ = diag(λ semidefinite λ1 ≥ · · · ≥ λm ≥ 0. Therefore, we can take Λ1/2 = √1 , ... , λm√) and since Θ is positive1/2 diag( λ1 , ... , λm ) and write Θ = V Λ Λ1/2 V t . Now denote Q0 = QV Λ1/2 , then: ΣX = QΘQ t + Ψ = QV Λ1/2 Λ1/2 V t Q t + Ψ = Q0 Q0t + Ψ. Thus ΣX = Q0 Q0t + Ψ, with Q0 ∈ Mp×m (R) and Ψ ∈ Mp (R), with Ψ diagonal and positive definite, so we are on the hypotheses of Theorem 2.3. Hence, there exists two random vectors f0 = (f01 , ... , f0m )t and u0 = (u01 , ... , u0p )t that satisfy the orthogonal factor model for X with loadings matrix Q0 and Cov(u0 ) = Ψ, that is, satisfying X = Q0 f0 + u0 , Cov(u0 ) = Ψ, E[u0 ] = 0p×1 , Cov(f0 ) = Im , E[f0 ] = 0m×1 and Cov(f0 , u0 ) = 0m×p . Now, define the random vectors f := V Λ1/2 f0 and u := u0 , and let’s see that these vectors give a solution to the generalized factor model for X with loadings matrix Q, Cov(f ) = Θ and Cov(u) = Ψ. It holds: X = Q0 f0 + u0 = QV Λ1/2 f0 + u0 = Qf + u and it also holds: Cov(u) = Cov(u0 ) = Ψ, E[u] = E[u0 ] = 0p×1 , Cov(f ) = Cov(V Λ1/2 f0 ) = V Λ1/2 Cov(f0 )(V Λ1/2 )t = V Λ1/2 Im Λ1/2 V t = Θ, E[f ] = E[V Λ1/2 f0 ] = 0m×1 , Cov(f , u) = Cov(V Λ1/2 f0 , u0 ) = 0m×p . Thus, X = Qf + u with f and u satisfying the conditions in Definition 3.1 and the proof is complete. In view of Theorem 3.3 and similarly to the orthogonal case, to fit the generalized factor model to a e , ΣX is replaced by the sample covariance matrix S and one tries to obtain parameters Q b ∈ data matrix X b ∈ Mm (R), Θ b symmetric and positive semidefinite, and Ψ b ∈ Mp (R), Ψ b diagonal Mp×m (R), with m < p, Θ and positive definite such that the equality bΘ bQ bt + Ψ b S =Q holds, at least approximately. Factor models such as the one discussed in this section are used in confirmatory factor analysis. In this technique the value of some parameters of the model is fixed in advance, and only the free parameters are estimated. For example, it is common to fix some loadings to be zero. Values are fixed to obtain a solution with a meaningful structure for the researcher. For this reason, if the prefixed model fits the observed data, rotations may not be necessary, in contrast to the exploratory case. It is usual to perform an exploratory analysis before the confirmatory one, in order to choose a model that is not in contradiction with the data, but the exploratory and confirmatory procedures should be checked on different subsamples to honestly confirm the model.

Expression of gratitude I would like to thank Xavier Bardina, editor of Reports@SCM, for seeing an interest in sharing the proofs given in this article, when I first exposed them in my final degree project. I would also like to thank Mercè Farré, my project advisor in the Math Department of Universitat Autònoma de Barcelona, for her precious comments about this document.

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References [1] T. Banachiewicz, “Zur Berechnung der Determinanten, wie auch der Inversen und zur darauf basierten Auflosung der Systeme linearer Gleichungen”, Acta Astronom. Ser. C 3 (1937), 41–67. [2] W.K. Härdle, L. Simar, “Applied Multivariate Statistical Analysis”, Springer, Berlin Heidelberg, 2012. [3] K.G. Jöreskog, “Some contributions to maximum likelihood factor analysis”, Psychometrika 32(4) (1967), 443–482.

[4] K.G. Jöreskog, A.S. Goldberger, “Factor analysis by generalized least squares”, Psychometrika 37(3) (1972), 243–260. [5] K.V. Mardia, J.T. Kent, J.M. Bibby, “Multivariate Analysis”, Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New YorkToronto, Ont., 1979. [6] J. Schur, “Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind”, J. Reine Angew. Math. 1917(147) (1917), 205–232.

Reports@SCM 6 (2021), 1–9; DOI:10.2436/20.2002.02.23.

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AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Hirzebruch signature theorem and exotic smooth structures on the 7-sphere ∗ Guifré

Sánchez Serra

Universitat Autònoma de Barcelona guifre.sanchez@gmail.com ∗Corresponding author

Resum (CAT) L’existència d’estructures diferencials no estàndards per a Sn no es va demostrar fins l’any 1956, quan J. Milnor va donar una construcció explı́cita d’una sèrie d’exemples pel cas n = 7, [4]. Fins llavors, s’assumia que no hi havia cap diferència fonamental entre esferes topològiques i esferes llises. El descobriment va suposar un punt d’inflexió en la topologia algebraica i de varietats, que continuaria amb la caracterització dels anomenats grups d’esferes homotòpiques, [2]. Un dels resultats que va fer possible la prova de Milnor va ser el teorema de la signatura de Hirzebruch, que dona una fórmula pel càlcul de la signatura d’una varietat (diferenciable) compacta i orientada. L’objectiu d’aquest treball és contextualitzar aquest teorema, aixı́ com mostrar el seu paper en la construcció de les primeres 7-esferes exòtiques.

Abstract (ENG) The existence of non-standard smooth structures on Sn was not proven until 1956, when J. Milnor presented an explicit construction for the case n = 7, [4]. Until then, it was assumed that there was no fundamental difference between topological and smooth spheres. This had profound implications in the field of manifold and algebraic topology, and was immediately endorsed by subsequent research, which lead to the characterization of the so called groups of homotopy spheres, [2]. One of the results that made Milnor’s approach possible was Hirzebruch’s signature theorem, which gives a formula to compute the signature of a (smooth) compact oriented manifold. The aim of this work is to contextualize this theorem, as well as to show its role in the construction of the first exotic 7-spheres.

Acknowledgement The author was partially supported by grant

Keywords: characteristic classes, manifold cobordism, exotic spheres. MSC (2010): 55R15, 57R20, 57R55. Received: July 16, 2021. Accepted: August 26, 2021. http://reportsascm.iec.cat

COLAB 2019 of the Spanish Ministry of Education and Professional Training. The author would also like to thank Dr. Wolfgang Pitsch for his invaluable advice throughout the development of this work.

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1. Classification of vector bundles An n-real vector bundle is a triplet (π, E , B), where E (total space) and B (base space) are topological spaces and π : E −→ B (projection) is a continuous map s.t. for each b ∈ B, π −1 (b) ∼ = Rn as real vector n −1 ∼ spaces, and there exists a neighbourhood U containing b s.t. U ×R = π (U) through a homeomorphism h that restricts to a linear isomorphism {b} × Rn −→ π −1 (b) for each b ∈ U. We refer to the pair (U, h) as a local coordinate system for ξ and we say that ξ is locally trivial. The notion of an induced bundle along a map is ubiquitous in the constructions that are to follow and will be used extensively throughout the work. Given a vector bundle ξ := (π, E , B), a topological space B 0 and a continuous map f : B 0 −→ B, one can define another vector bundle, f ∗ ξ := (π 0 , E 0 , B 0 ), with E 0 := {(b, e) ∈ B 0 × E | π(e) = f (b)} and π 0 : E 0 −→ B 0 , (b, e) 7−→ b. The vector space structure in the fibers, (π 0 )−1 (b), is defined by λ1 (b, e1 ) + λ2 (b, e2 ) = (b, λ1 e1 + λe2 ). We refer to f ∗ ξ as the induced bundle or pullback bundle of ξ by f . One can show that f ∗ ξ is locally trivial by expressing its local coordinate systems in terms of local coordinate systems for ξ. The reader is referred to [5, §3] for a more detailed description. One of the most important results regarding real vector bundles is the following classification theorem (see [5, §2–5], [1, §1]), which translates the problem of classifying isomorphic vector bundles into a homotopy problem: Theorem 1.1. Let Fn (B) be the set of n-vector bundles over a paracompact Hausdorff base B modulo isomorphism, and let [B, Gn ] be the set of homotopy classes of maps from B to Gn . Then, the map: Φ : [B, Gn ] −→ Fn (B) [f ] 7−→ [f ∗ γ n ]

(1)

is a bijection; where γ n is the universal bundle over Gn (R∞ ). To prove this theorem we need three important results. We will explain each of them and briefly detail their proofs. The first one is necessary to ensure that Φ is well defined: Proposition 1.2. Let ξ be a vector bundle with projection π : E −→ B, and let f , g : C −→ B be continuous maps, with C paracompact1 . If f and g are homotopic, then f ∗ ξ ∼ = g ∗ ξ. Sketch of proof. The proof of Proposition 1.2 is based on the fact that every n-vector bundle over the interval [0, 1] =: I is trivial. From this, one then shows that, given a vector bundle on C , there is an open cover such that for each of its open sets Ui the restriction of the bundle to Ui × I is trivial. In particular, on each of these open sets, the bundles at each end are clearly isomorphic. One then glues these local isomorphisms to show that, given i0 , i1 : C −→ C × I , ik (c) = (c, k), k = 0, 1, and a vector bundle ξ over C × I , the induced bundles by i0 and i1 over C are isomorphic, which allows us to complete de proof. Indeed, given f , g : C −→ B homotopic, and h : C × I −→ B a homotopy s.t. h0 = f , h1 = g , we have f = h◦i0 , g = h◦i1 , which implies f ∗ ξ ∼ = i0∗ (h∗ ξ), g ∗ ξ ∼ = i1∗ (h∗ ξ). This yields f ∗ ξ ∼ = g ∗ ξ, by the observation on the induced bundles by i0 and i1 . 1

Paracompactness (every open cover admits a locally finite refinement) will be necessary to ensure we can use partitions of unity arguments.

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The second of the aforementioned results is necessary to ensure that Φ is surjective: Proposition 1.3. Any n-vector bundle over a paracompact Hausdorff base B admits a bundle morphism f : ξ −→ γ n . Sketch of proof. This result is an extension of its “finite” counterpart, which ensures the existence of a bundle morphism ξ −→ γkn (for a sufficiently large k), with γkn the canonical vector bundle over the Grassmannian Gn (Rn+k ) (following the notation in [5]). The latter assumes B compact and Hausdorff, which allows us to take a finite covering {U1 , ... , Ur } s.t. ξ | Ui is trivial for each i. Thus, each Ui admits n linearly independent sections. By using partitions of unity, one can extend these sections over B, and find a finite number of them, s1 , ... , sn , ... , sn+k , s.t. for every b ∈ B, {s1 (b), ... , sn+k (b)} generates Fb (ξ). Thus, the map gb : (t1 , ... , tn+k ) 7−→ Σti si (b) between Rn+k and Fb (ξ) is surjective for each b. Denoting Vb := (ker gb )⊥ and fb the linear isomorphism given by gb | Vb , it is clear that the map f : e 7−→ (Vb , fb−1 (e)) defines an isomorphism between ξ and γkn . The proof for the γ n case works similarly, letting k go to infinity and taking into account that the Vb ’s are embedded in R∞ , which allows to weaken the condition on the base, B (hence the paracompactness). The last of these three core results related to Theorem 1.1 proves that Φ is injective: Proposition 1.4. Let ξ be an n-vector bundle over a paracompact Hausdorff base B, and let f , g : ξ −→ γ n be bundle morphisms. Then, f and g are homotopic. Sketch of proof. From the proof of Proposition 1.3 it is clear that any morphism f : ξ −→ γ n is of the form e 7−→ (f˜ (fiber over e),f˜(e)), where f˜ is a continuous map between E (ξ) and R∞ that is linear injective over the fibers of ξ. Let f˜, g̃ be these maps for f and g respectively. Two scenarios are distinguished, based on the relation between f˜ and g̃ . Firstly, we assume f˜(e) 6= λg̃ (e) for any λ < 0, with e ∈ E (ξ). Then, we can easily define a bundle homotopy h : E (ξ) × [0, 1] −→ E (γ n ) between f and g of the form (e, t) 7−→ (h̃t (fiber over e), h̃t (e)), by setting h̃t (e) := (1 − t)f˜(e) + t g̃ (e). The condition on f˜ and g̃ allows to prove the injectivity of h̃t , and the fact that f and g are both bundle maps, ensures continuity and linearity, which proves ht is a morphism for each t. Moreover, it can also be seen that h is continuous, by inspecting its restriction to the bases. This proves h is a homotopy. Secondly, we make no assumption on f˜ and g̃ . To prove that f and g are homotopic, we consider the maps d̃1 , d̃2 : R∞ −→ R∞ s.t. d̃1 (ei ) = e2i−1 , d̃2 (ei ) = e2i , for i = 1, 2, 3, ... , where the ej ’s are the canonical base vectors for R∞ . These maps induce two morphisms d1 , d2 from γ n to itself, that we can use to obtain r1 := d1 ◦ f and r2 := d2 ◦ g . It is clear that f˜(e) 6= λr̃1 (e), with λ < 0, for all e ∈ E (ξ), which implies f ' r1 . Analogously, g ' r2 . We can verify that the same condition holds between r̃1 and r̃2 . Thus, f ' g , as desired. With these results, we can prove Theorem 1.1: Proof of Theorem 1.1. Proposition 1.2 guarantees that Φ is a well defined map between [B, Gn ] and Fn (B). We can see that Φ is injective using Proposition 1.4. Indeed, [f ∗ γ n ] = [g ∗ γ n ] implies that there exists an isomorphism φ : E (f ∗ γ n ) −→ E (g ∗ γ n ). Composing with ĝ : E (g ∗ γ n ) −→ E (γ n ) we obtain a bundle morphism ϕ := ĝ ◦ φ with induced base map ϕ = g ◦ φ = g , given that φ = id, since φ is an isomorphism. By Proposition 1.4, ϕ and fˆ are homotopic bundle morphisms, which yields [ϕ] = [g ] = [f ], as desired. We now prove that Φ is surjective. Let [ξ] ∈ Fn (B). By Proposition 1.3 we know there exists a bundle morphism f : ξ −→ γ n of the form f (e) = (f˜ (fiber over e), f˜(e)), with f˜ : E (ξ) −→ R∞ continuous and

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linear injective at the level of the fibers. This provides us with a homotopy class [f ] where f is the base ∗ induced map associated to f , i.e. b 7−→ f˜(Fb (ξ)). We observe that f γ n and ξ are isomorphic, since f is a morphism. Thus, Φ([f ]) = [ξ], as desired. An n-complex vector bundle ω is defined similarly, with all R-linear objects and properties substituted by their C-linear analogues. We also have a classification theorem for these bundles, similar to Theorem 1.1. However, complex vector bundles present a much richer structure than their real counterparts. On one hand, every complex vector bundle can be thought as a real vector bundle, by omitting its complex structure (we can just think of its fibers as real vector spaces of twice the original dimension). We refer to this bundle as the underlying real vector bundle of ω, and denote it by ωR . On the other hand, we can define the conjugate bundle of ω, ω, as the complex vector bundle with same underlying bundle, i.e. ω R = ωR , but with opposite complex structure, i.e. id : E (ω) −→ E (ω) is C-conjugate linear on fibers.

2. Characteristic classes Theorem 1.1 is key to define characteristic classes for vector bundles. Let c be a cohomology class in H i (Gn ; R), with R some coefficient ring, and let ξ be a certain n-vector bundle with (paracompact Haus∗ dorff) base B. From Theorem 1.1 there is a unique homotopy class [f ξ ] ∈ [B, Gn ] s.t. f ξ γ n ∼ = ξ. ∗ Let f ξ : H i (Gn ; R) −→ H i (B; R) be the induced morphism between cohomology groups, and define2 ∗ c(ξ) := f ξ c. This cohomology class in H i (B; R) is the characteristic class of ξ determined by c. From this construction we make two important observations. First, the correspondence ξ 7−→ c(ξ) is natural w.r.t. bundle morphisms, meaning that if we have g : B(ξ) −→ B(η) covered by a bundle morphism, then c(ξ) = g ∗ c(η). This can be seen by noting that f η ◦ g and f ξ are homotopic. Since g is covered by a bundle morphism, ξ ∼ = g ∗ η, which allows us to write c(g ∗ η) = g ∗ c(η). Second, we note that given any ∗ “natural” correspondence ξ 7−→ c(ξ), we will necessarily have c(ξ) = f ξ c(γ n ), since the map f ξ is always covered by a bundle morphism between ξ and γ n . So this construction is as general as one can ask for and tells us that the ring of characteristic cohomology classes of n-vector bundles over paracompact Hausdorff base, with coefficients in R, is canonically isomorphic to H ∗ (Gn ; R). Thus, the computation of H ∗ (Gn ; R) is relevant to the study of fundamental properties of n-vector bundles over paracompact Hausdorff base. We will briefly present the most important 3 types of characteristic classes: the Stiefel–Whitney classes, the Chern classes and the Pontrjagin classes. From now on, we assume that all vector bundles have paracompact Hausdorff bases, unless otherwise stated. The Stiefel–Whitney classes are characteristic classes of non oriented vector bundles, that are defined in Z/2 cohomology groups. The existence and uniqueness of such classes is difficult to prove, and is based on the computation of H ∗ (Gn ; Z/2), which can be found, for example, in [5, §6–8]. Hence, most often we find these classes defined axiomatically as follows: Definition 2.1 (Stiefel–Whitney classes. Axiomatic definition). Axiom I. Given a vector bundle ξ, there is a unique sequence of characteristic classes wi (ξ) ∈ H i (B(ξ); Z/2), i = 0, 1, 2, ... , that we refer to as the Stiefel–Whitney classes of ξ. Moreover, w0 (ξ) = 1 and if ξ is an n-vector bundle, wi (ξ) = 0 for all i > n. 2

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Note that this is well defined, because the induced morphisms from two maps sharing homotopy type are equal.

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Axiom II. Naturality If f : B(ξ) −→ B(η) is covered by a bundle morphism between ξ and η, then: wi (ξ) = f ∗ wi (η), i = 0, 1, 2 ... , where f ∗ is the induced morphism between cohomology groups over Z/2. Axiom III. Let ξ and η be two vector bundles over the same base, B, then: wk (ξ ⊕ η) =

k X

wi (ξ) ^ wk−i (η),

k = 0, 1, 2, ... ,

(2)

i=0

where ^ denotes the cup product between cohomology classes. Axiom IV. w1 (γ11 ) 6= 0, where γ11 is the canonical line bundle over P1 . We define the total Stiefel–Whitney classQw (ξ), for an n-vector bundle ξ, as the formal sum w0 (ξ) + w1 (ξ) + · · · + wn (ξ) + 0 + · · · in the ring H (B(ξ); Z/2) of series of the form a0 + a1 + a2 + · · · , with each ai ∈ H i (B(ξ); Z/2). This notation serves to express more synthetically Axiom III from Definition 2.1, which can now be written as w (ξ ⊕ η) = w (ξ)w (η). Some basic properties of the Stiefel–Whitney classes are: Proposition 2.2. (i) The Stiefel–Whitney classes of two isomorphic bundles are equal. (ii) Let ε be the trivial bundle, then wi (ε) = 0 for i > 0. (iii) Let η be a vector bundle and ε the trivial bundle over the same base, then wi (ε ⊕ η) = wi (η) for i = 0, 1, 2, ... (iv) Let ξ be an euclidian n-vector bundle with a non-zero section s : B −→ E ; then wn (ξ) = 0. More generally, if ξ has k linearly independent sections, then wn−k+1 (ξ) = · · · = wn (ξ) = 0. From the axioms in Definition 2.1 we can compute the total Stiefel–Whitney class of the canonical line bundle over the real projective space Pn , γn1 : Proposition 2.3. w (γn1 ) = 1 + a, where a is the generator of H 1 (Pn ; Z/2). Proof. Let j : P1 −→ Pn be the inclusion [x] 7−→ [i(x)], with i the canonical inclusion R2 ,−−→ Rn+1 . The map j is covered by the bundle morphism J between γ11 and γn1 that sends ([x], v ) ∈ E (γ11 ) to (j([x]), i(v )) ∈ E (γn1 ). By Axioms II and IV, we then have j ∗ wi (γn1 ) = wi (γ11 ) and j ∗ w1 (γn1 ) 6= 0, respectively. Since j ∗ is a morphism, w1 (γn1 ) 6= 0 necessarily, and by the structure of the cohomology ring H ∗ (Pn ; Z/2), w1 (γn1 ) = a, with a its generator. Finally, since dim γn1 = 1, wi (γn1 ) = 0 for i > 1 and w0 (γn1 ) = 1, by Axiom I. Thus, w (γn1 ) = 1 + a, which concludes the proof. Proposition 2.4. The total Stiefel–Whitney class of T Pn is (1 + a)n+1 , where a ∈ H 1 (Pn ; Z/2) is the ring generator of H ∗ (Pn ; Z/2). Proof. Assume that T Pn is isomorphic to Hom(γn1 , γ ⊥ ), where γ ⊥ is the orthogonal complement of γn1 1 1 in εn+1 Pn . Note then that Hom(γn , γn ) is a trivial bundle of dimension 1, which implies: T Pn ⊕ ε1 ∼ = Hom(γn1 , γ ⊥ ) ⊕ Hom(γn1 , γn1 ) ∼ = Hom(γn1 , γn1 ⊕ γ ⊥ ) ∼ = Hom(γ 1 , εn+1 ) ∼ = Hom(γ 1 , ε1 ⊕ · · · ⊕ ε1 ) ∼ =

n Hom(γn1 , ε1 )

n

(3)

⊕ · · · ⊕ Hom(γn1 , ε1 ) ∼ = γn1 ⊕ · · · ⊕ γn1 ,

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where we used Hom(γn1 , ε1 ) ∼ = γn1 and γn1 ⊕ γ ⊥ ∼ = εn+1 , by construction. Thus, using property (iii) from Proposition 2.2, Axiom III and Proposition 2.3 it is clear that: w (T Pn ) = w (T Pn ⊕ ε1 ) = w (γn1 ⊕ · · · ⊕ γn1 ) = w (γn1 )n+1 = (1 + a)n+1 .

(4)

We now prove T Pn ∼ = Hom(γn1 , γ ⊥ ). Let q : Sn −→ Pn be the quotient map. Note that Tx q(v ) = T−x q(−v ), which can be derived from the fact that q(x) = q(−x). Since q is a local diffeomorphism, Tx q is a linear isomorphism for each x. This allows identifying T Pn with the pairs {(x, v ), (−x, −v )} s.t. kxk = 1 and hx, v i = 0 through {(x, v ), (−x, −v )} 7−→ ([x], Tx q(v )). Observe that each of these pairs defines a linear map from hxi = F[x] (γn1 ) to hxi⊥ = F[x] (γ ⊥ ), determined by x 7−→ v . We can identify this map with the corresponding element in F[x] (Hom(γn1 , γ ⊥ )). This clearly allows defining a bundle isomorphism between T Pn and Hom(γn1 , γ ⊥ ): ϕ : T Pn −→ E (Hom(γn1 , γ ⊥ )) ([y ], u) ∼ {(x, v ), (−x, −v )} 7−→ L[y ],u : F[y ] (γn1 ) −→ F[y ] (γ ⊥ ), x 7−→ v with x =

y ky k

(5)

and v = Tx q −1 (u).

Chern classes are characteristic classes associated to complex vector bundles. They are constructed as in the real case, but using the corresponding classification theorem, and are cohomology classes over Z. We provide, as for the Stiefel–Whitney classes, an axiomatic definition: Definition 2.5 (Chern classes. Axiomatic definition). Axiom I. Given a complex vector bundle ω, there is a unique sequence of characteristic classes ci (ω) ∈ H 2i (B(ξ); Z), i = 0, 1, 2, ... , that we refer to as the Chern classes of ω. Moreover, c0 (ω) = 1 and if ω is an n-complex vector bundle, ci (ω) = 0 for all i > n. Axiom II. Naturality. If f : B(ω) −→ B(ω 0 ) is covered by a bundle morphism between ω and ω 0 , then: ci (ω) = f ∗ ci (ω 0 ), i = 0, 1, 2 ... Axiom III. Let ω and ω 0 be two complex vector bundles over the same base, B, then: ck (ω ⊕ ω 0 ) =

k X

ci (ω) ^ ck−i (ω 0 ),

k = 0, 1, 2, ...

(6)

i=0

Axiom IV. c1 (γ11 ) 6= 0, where γ11 is the canonical line bundle over CP1 . A total Chern class c(ω) is defined for complex vector bundles in the same way as in the Stiefel– Whitney case. Chern classes share properties (i)–(iii) from Proposition 2.2 with the Stiefel–Whitney classes. However, there is a distinctive feature that we need to present for future developments: Proposition 2.6. Given a complex vector bundle ω, ck (ω) = (−1)k ck (ω), k ≥ 0, where ω is the conjugate bundle of ω. Proposition 2.7. The total Chern class of T CPn is (1 + t)n+1 , where t = −c1 (γn1 ), and γn1 is the canonical line bundle over CPn .

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Sketch of proof. We can easily follow the steps in Proposition 2.4, to show that: n+1) n+1) T CPn ⊕ ε1 ∼ = Hom(γn1 , ε1 ) ⊕ · · · ⊕ Hom(γn1 , ε1 ) ∼ = γ 1n ⊕ · · · ⊕ γ 1n ,

(7)

where we have used that the dual of a complex vector bundle equipped with an hermitian metric is canonically isomorphic to its conjugate bundle. This can be seen by noting that the map v 7−→ h·, v i between E (ω) and E (Hom(ω, ε1 )) is a (complex) bundle isomorphism. Thus, using Proposition 2.6: c(T CPn ) = c(γ 1n )n+1 = (1 − c1 (γn1 ))n+1 = (1 + t)n+1 .

(8)

Pontrjagin classes are characteristic cohomology classes over Z for real, possibly oriented, vector bundles. They can be thought as the oriented analogues of the Stiefel–Whitney classes, as they allow to distinguish between different vector bundle orientations. They are defined through Chern classes as follows: pi (ξ) := (−1)i c2i (ξ ⊗ C) ∈ H 4i (B(ξ); Z),

i = 0, 1, 2, ... ,

(9)

where ξ is a real vector bundle of dimension n, and ξ ⊗ C its complexification, that is, the complex vector bundle over the same base whose fibers are the products Fb (ξ) ⊗R C (treating C as a vector space over R). By Axiom I of the Chern classes, it is clear that pi (ξ) = 0 for all i > n/2. Hence, the total Pontrjagin class of ξ can be written as: p(ξ) := 1 + p1 (ξ) + p2 (ξ) + · · · + p n (ξ). 2

As they are derived from Chern classes, Pontrjagin classes also share properties (i)–(iii) from Proposition 2.2. However, they exhibit two more properties: Proposition 2.8. (i) Given two real vector bundles ξ, η, p(ξ ⊕ η) = p(ξ)p(η) modulo order 2 terms. (ii) Let ω be an n-complex vector bundle. Then, the Chern classes of ω determine the Pontrjagin classes of ωR , through the following relation: pk (ωR ) = ck (ω)2 − 2ck−1 (ω)ck+1 (ω) + · · · + (−1)k 2c2k (ω).

(10)

Alternatively, we can write: 1 − p1 (ωR ) + p2 (ωR ) − · · · + (−1)n pn (ωR ) = c(ω)c(ω) modulo order 2 terms.

(11)

Proposition 2.8 is, essentially, a consequence of Proposition 2.6 combined with the following bundle isomorphisms: ξ ⊗ C ∼ = ξ ⊗ C, with ξ a real vector bundle (property (i)); and ωR ⊗ C ∼ = ω ⊕ ω, with ω an n-complex vector bundle (property (ii)). 2 Proposition 2.9. The total Pontrjagin class of the underlying vector bundle of T CPn is 1 + n+1 1 t + n n+1 4 n+1 t 2 2 , where t = −c1 (γ 1 ). Alternatively, p(T CPn ) = (1 + t 2 )n+1 . t + · · · + n n R 2 2

Proof. Denote τ := T CPn . From Propositions 2.7 and 2.8(ii), it is clear that: c(τ )c(τ ) = (1 + t)n+1 (1 − t)n+1 = (1 − t 2 )n+1 = 1 − p1 (τR ) + p2 (τR ) − · · · + (−1)n pn (τR )

(12)

modulo order 2 terms. Taking into account that t ∈ H 2 (CPn ; Z) and comparing elements with same 2k dimension in cohomology, we find pk (τR ) = n+1 k t , k = 0, 1, ... , n, which allows us to write p(τR ) = (1 + t 2 )n+1 . Finally, using H i (CPn ; Z) ∼ = 0 for i > 2n, we obtain the result.

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3. Hirzebruch signature theorem Manifold cobordism is a well known equivalence relation between closed differentiable manifolds. Given two closed differentiable n-manifolds (n-manifolds from now on) M1 , M2 , we say they belong to the same cobordism class, or that they are cobordant, and denote it M 1 = M 2 , if M1 t M2 is the boundary of a compact (n + 1)-manifold, N, that we refer to as a cobordism between M1 and M2 . Similarly, two closed n-manifolds belong to the same oriented cobordism class if M1 t (−M2 ) is the boundary of a compact oriented (n + 1)-manifold (through an orientation preserving diffeomorphism). These are equivalence relations (see [5, 3]) over closed manifolds and closed oriented manifolds respectively. We can provide a brief explanation to justify why this may be true for the oriented case: M is cobordant to itself, because M t (−M) = ∂(M × [0, 1]); if W is a cobordism between M1 and M2 , it is clear that −W is a cobordism between M2 and M1 ; finally, given W1 , W2 cobordisms between M1 , N and N, M2 respectively, we have that W1 t W2 /∼, conveniently identifying N with −N (using the collar neighborhood theorem), is a cobordism between M1 and M2 . One of the major contributors to cobordism theory was R. Thom, who helped establishing its foundations, but also gave birth to some of its most important results. The classification of closed (oriented) manifolds up to cobordism was a consequence of the efforts by R. Thom, L. Pontrjagin and C. T. C. Wall, who were able to connect characteristic classes with cobordism classes in a brilliant manner. To present this result, we need to introduce the notion of characteristic numbers: Definition 3.1 (Stiefel–Whitney numbers and Pontrjagin numbers). Let M be a closed m-manifold and N a closed oriented 4n-manifold. Let I = (i1 , ... , ir ) and J = (j1 , ... , js ) be partitions of m and n respectively, and define: wI [M] := hwi1 (TM) · · · wir (TM), µi,

pJ [N] := hpj1 (TN) · · · pjs (TN), νi,

(13)

where µ ∈ Hm (M; Z/2) and ν ∈ H4n (N; Z) are the fundamental homology classes of M and N. The numbers wI [M] ∈ Z/2 and pI [N] ∈ Z are the Stiefel–Whitney numbers of M and the Pontrjagin numbers of N, respectively. If dim N 6≡ 0 (4), we say that the Pontrjagin numbers of N vanish. Theorem 3.2 (Classification of closed oriented manifolds modulo oriented cobordism). Stiefel–Whitney and Pontrjagin numbers completely classify closed oriented manifolds modulo oriented cobordism. Thus, given two closed oriented n-manifolds M1 and M2 , they belong to the same oriented cobordism class if and only if wI [M1 ] = wI [M2 ] and pJ [M1 ] = pJ [M2 ] for all partitions I , J of n and n4 , respectively. Both oriented and unoriented cobordisms give rise to a group structure between cobordism classes of closed manifolds of the same dimension, by means of the disjoint union. Since, we are most interested in the oriented case, we present these groups and the graded ring they form, ΩSO ? , which is an important object in the subsequent discussion. Definition 3.3 (Oriented cobordism groups and oriented cobordism ring). We define the oriented cobordism group of dimension n as the set ΩSO n := {M | M closed oriented n-manifold} together with the following operation: M + N := M t N. (ΩSO n , +) is an abelian group with identity ∅. L∞ SO We define the oriented cobordism ring as the graded commutative ring ΩSO := ? n=0 Ωn , with component-wise sum and product given by the cartesian product between manifolds: M 1 ×M 2 := M1 × M2 .

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To prove that the sum of oriented cobordism classes is well defined one must see that if W = W1 t W2 , with W1 , W1 , W compact oriented (n + 1)-manifolds, then ∂W = ∂W1 t ∂W2 . On the other hand, given a cobordism W between M1 , M2 , and given a closed oriented manifold N, W × N is a cobordism between M1 × N and M2 × N. Thus, with Mi = Ni , i = 1, 2, M1 × M2 = N1 × M2 = N1 × N2 , which proves the product is also well defined. Note also that ΩSO ? is commutative in the graded sense, because M1 × M2 and (−1)dim M1 dim M2 M2 × M1 are diffeomorphic as oriented manifolds. The structure of ΩSO ? was thoroughly studied by R. Thom, whose findings can be summarized as follows (see [5, §16–17]): Theorem 3.4. (i) ΩSO n is a finite group for n ≡ 0 (4) and is finitely generated with rank p(k), the number of partitions of k, when n = 4k. Moreover, in the case where n = 4k, the products CP2i1 × · · · × CP2ir , with I = (i1 , ... , ir ) partition of k, are a set of independent generators. 2 4 6 (ii) ΩSO ? ⊗ Q is a polynomial ring over Q generated by CP , CP , CP , ...

Note that (ii) is a direct consequence of (i). Indeed, since the Z-module product ΩSO ? ⊗ Q is effectively SO , k ≥ 0; given that these , we are left, by (i), with the groups Ω eliminating the torsion elements in ΩSO ? 4k are generated by the products CP2i1 ×· · ·×CP2ir , the results follows. Note also that in ΩSO ? ⊗Q all products are commutative, given that dim CP2i ≡ 0 (2). An important cobordism invariant, other than Pontrjagin and Stiefel–Whitney numbers, is the signature. In fact, this section is devoted to Hirzebruch’s signature theorem, which presents a formula for the computation of the signature of a manifold, in terms of its Pontrjagin numbers. Definition 3.5 (Signature). The signature of a compact oriented manifold M of dimension 4k, is the signature of the rational bilinear symmetric form: H 2k (M; Q) × H 2k (M; Q) −→ Q (u, v ) 7−→ hu ^ v , µi,

(14)

where µ is the fundamental homology class of M over Q, consistent with its orientation. We denote it by σ(M). If dim M 6≡ 0 (4), we define σ(M) := 0. From now on, given a manifold M, we will write simply M to refer to its cobordism class. Lemma 3.6. The map M 7−→ σ(M) determines an algebra homomorphism between ΩSO ? ⊗ Q and Q. Lemma 3.6 is a consequence of three important properties of the signature: σ is an additive and multiplicative function with respect to the corresponding operations between cobordism classes, i.e. σ(M1 + M2 ) = σ(M1 ) + σ(M2 ) and σ(M1 × M2 ) = σ(M1 )σ(M2 ); and σ is a cobordism invariant, that is, if M1 and M2 belong to the same cobordism class, then σ(M1 ) = σ(M2 ) (alternatively σ(M) = 0 if M = ∅). To establish Hirzebruch’s theorem it is necessary to introduce the notion of multiplicative sequences: L∞ ? Definition 3.7. Let R be a commutative i=0 Ai be a graded commutative Q ring with unity and let A := (in the classical sense) R-algebra. Let A be the set of formal series a0 +a1 +a2 +· · · , with each ai ∈ Ai , and Q Q define A1 := {a ∈ A | a0 = 1}. Let {Kn (x1 , ... , xn )}n≥1 be a sequence of polynomials with coefficients Q in R s.t. Kn (x1 , ... , xn ) is homogoneous of degree n with dim xi = i for each i. Given a ∈ A1 , define

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Q

K (a) := 1 + K1 (a1 ) + K2 (a1 , a2 ) + · · · ∈ A1 . We say that {Kn }n≥1 is a multiplicative R-sequence, if, for Q ? any R-algebra A of the form described, K (ab) = K (a)K (b) for all a, b ∈ A1 . We may write Ki (a) to denote Ki (a1 , ... , ai ), with K0 (a) = 1. An important result regarding multiplicative sequences is the following lemma (see [5, p. 221]): Lemma 3.8. Let R be a commutative ring with unity, and let f (t) = 1 + r1 t + r2 t 2 + · · · be a formal power series with coefficients in R. Then, there exists a unique multiplicative R-sequence {Kn } s.t. K (1+t) = f (t). We refer to K as the multiplicative sequence associated to f (t). Another important cobordism invariant, arising from multiplicative sequences, and with similar properties as the signature, is the K -genus: Definition 3.9. Let M be a compact oriented 4n-manifold, and let {Kn } =: K be a multiplicative Q-sequence. We define the K -genus of M, K [M], as the number hKn (p1 , ... , pn ), µ4n i, where pj := pj (TM) for each j, and µ4n is the fundamental class of M. If the dimension of M is not a multiple of 4, K [M] := 0. Note that the K -genus is just a rational combination of some Pontrjagin numbers of M. Since Pontrjagin numbers are additive under cobordism sum (which can be easily proven using that they are cobordism invariants and also H i (M1 t M2 ) ∼ = H i (M1 ) ⊕ H i (M2 )), it is clear that M 7−→ K [M] is also additive. It can also be seen that K [M1 × M2 ] = K [M1 ]K [M2 ], using a much harder result from homological algebra known as Künneth theorem. Thus, similar to the case of the signature, we have that: Lemma 3.10. Given a multiplicative Q-sequence K , the map M 7−→ K [M] determines an algebra homomorphism between ΩSO ? ⊗ Q and Q. Theorem 3.11 (Hirzebruch’s signature theorem [5, §19]). Let L := {Ln } be the multiplicative sequence associated to the power series: √

f (x) =

x 1 1 (−1)n−1 22n Bn x n √ = 1 + x − x2 + · · · + + ··· 3 45 (2n)! tanh x

(15)

Then, the signature, σ(M), of a compact oriented manifold M, equals the L-genus of M, L[M]. Hence, for a compact oriented 4n-manifold M, we have: σ(M) = hLn (p1 , ... , pn ), µ4n i.

(16)

Proof. By Lemmas 3.6 and 3.10, both the signature and the L-genus define homomorphisms between ΩSO ? ⊗ SO Q and Q. Thus, it suffices to prove the result for a set of generators of Ω? ⊗ Q. Using Theorem 3.4(ii), it is clear that we only have to show that σ(CP2n ) = L[CP2n ], n ≥ 1. Since H 2n (CP2n ; Q) is generated by t n , with t = −c1 (γn1 ), it is clear that σ(CP2n ) = ht 2n , µ4n i. On the other hand, by Proposition 2.9, the total Pontrjagin class of CP2n is (1 + t 2 )2n+1 . Now, since L is the multiplicative sequence associated to √ √ 3.8: L((1 + t 2 )2n+1 ) = L(1 + t 2 )2n+1 = (t/ tanh t)2n+1 , where we use f (x) = x/ tanh x, by Lemma L ∞ dim t 2 = 1 in the Q-algebra i=0 H 4i (CP2n ; Q). Thus: L[CP2n ] = hLn (p1 , ... , pn ), µ4n i =

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2n+1 t , µ 4n tanh t

= C ht 2n , µ4n i = C σ(CP2n ),

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Guifré Sánchez Serra

where C is the coefficient of t 2n in the power series of (t/ tanh t)2n+1 . Note that C coincides with the −1 degree coefficient of the power series around 0 of the complex function 1/(tanh z)2n+1 . Hence, by the residue theorem: I 1 1 C= dz, (18) 2πi γ (tanh z)2n+1 with γ a sufficiently small, positively oriented, regular closed path around the origin. Consider now the substitution u := tanh z, with dz = du/(1 − u 2 ), and note that: I I 1 1 du 1 1 + u2 + u4 + · · · = du = +1, (19) C= 2πi γ 0 u 2n+1 1 − u 2 2πi γ 0 u 2n+1 where we are again using the residue theorem. This concludes the proof, since L[CP2n ] = C σ(CP2n ).

4. Exotic structures One of the major mathematical achievements of the second half of the 20th century was the discovery of topological spheres that were not diffeomorphic to the standard sphere. The first examples, the exotic 7-spheres, were unveiled by J. Milnor in 1956 (see [4]). This had profound implications in the field of manifold and algebraic topology, since, until Milnor’s paper, a fundamental difference between topological and differentiable spheres was not expected. The purpose of this section is to introduce Milnor’s construction and to link what has been exposed so far with the existence of non-standard smooth structures in S7 . Milnor uses G -bundles, which are based on the concept of vector bundles, but allow fibers to be arbitrary topological spaces, connected through transition functions of the form (b, x) 7−→ (b, gij (b)x), where g : Ui ∩ Uj −→ G is a continuous map and G is a subgroup of homeomorphisms from F (the base fiber) to itself. For G -bundles over Sn , there is a classification theorem, analogous to the one presented for vector bundles, which establishes a one-to-one correspondence between bundle isomorphism classes and homotopy classes in πn−1 (G ). As π3 (SO(4)) ∼ = Z ⊕ Z (see [7]), one can parametrize SO(4)-bundles over S4 with fiber S3 by two integers, and denote a set of representatives by ζhj , h, j ∈ Z. One can also write for the corresponding total spaces, E (ζhj ) =: Mhj , an explicit covering set of local charts, together with their transition functions. This allows Milnor to apply a deep result from Reeb (see [6, p. 11]) to show that if h + j = 1 and h − j = k, with k odd, then: Mk := M 1+k 1−k is homeomorphic to S7 . 2 , 2

Let now M be a 7-manifold, oriented by µ ∈ H7 (M), s.t. H 3 (M) = H 4 (M) = 0 (with coefficients in Z from now on). Let B be a 8-manifold s.t. ∂B and M are diffeomorphic through an orientation-preserving diffeomorphism. Then, from the long exact sequence of the pair (B, M) we deduce that j ? : H 4 (B, M) −→ H 4 (B) is an isomorphism, which allows us to define: q(B) := h(j ? )−1 (p1 (TB))2 , νi, where ν ∈ H8 (B, M) is the fundamental class of the pair (B, M), compatible with the orientation of M, i.e. ∂ν = µ. Define also τ (B) as the signature of the quadratic form over H 4 (B, M)/torsion, given by α 7−→ hα2 , νi. Under these conditions, the following is true: Proposition 4.1. The residue of 2q(B) − τ (B) modulo 7 is independent of B. This is a direct consequence of Hirzebruch’s signature theorem, and is essential to prove the existence of a non-standard smooth structure for S7 .

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Sketch of proof (Proposition 4.1). Let B1 , B2 be two 8-manifolds s.t. ∂B1 = ∂B2 = M. Define C := B1 t (−B2 )/∼, where ∼ identifies ∂B1 with ∂(−B2 ). Since C is a closed 8-manifold, we can apply Hirzebruch’s signature theorem to see that:

1 σ(C ) = 45 (7p2 (C ) − p12 (C )), ν , (20) where ν is an orientation for C compatible with the corresponding orientations, ν1 and −ν2 , for B1 and −B2 respectively. This implies 2hp1 (C )2 , νi − σ(C ) ≡ 0 (7). It is not difficult to see, through some homological algebra computations and using the aforementioned conditions, that the quadratic form associated to C , over H 4 (C ), is the direct sum of the quadratic forms over H 4 (B1 , M) and H 4 (B2 , M), as defined above, reversing the sign of the latter. This clearly implies σ(C ) = τ (B1 ) − τ (B2 ) and, similarly, hp1 (C )2 , νi = q(B1 ) − q(B2 ), which proves the statement. Thus, under the stated conditions, we can define λ(M) := 2q(B)−τ (B) ∈ Z/7. This invariant provides a simple criterion to determine whether or not M and S7 can be diffeomorphic: Proposition 4.2. If λ(M) 6= 0, M cannot be diffeomorphic to the boundary of an 8-manifold B with H 4 (B) = 0. In particular, if λ(M) 6= 0, M and S7 are not diffeomorphic. Since H 4 (B) ∼ = H 4 (B, M), the quadratic form over H 4 (B, M)/torsion is 0, which yields τ (B) = q(B) = 0 and, consequently, λ(M) = 0. If M and S7 are diffeomorphic, we can choose B = D8 and use H 4 (D8 ) = 0 to conclude. Let now M be one of the aforementioned Mhj . It is clear that the total space, Nhj , of the SO(4)-bundle, ηhj , that results from substituting the S3 fibers in ζhj by 4-disks, D4 , satisfies ∂Nhj = Mhj . A rather complex computation shows that p1 (TNhj ) = c(h − j)β, where c ∈ Z and β is a generator of H 4 (Nhj ) ∼ = Z. This can be further used, for the spaces Mk , to show that q(Nk ) = δc 2 k 2 , with δ = ±1. Finally, noting that τ (Nk ) = δ, it is clear that λ(Mk ) ≡ δ(2c 2 k 2 − 1) (7). Since the squares in Z/7 are 0, 1, 2, 4, we may choose k = 0, 3, 1, 5 to obtain λ(Mk ) 6= 0. This proves the following: Theorem 4.3. S7 admits at least one non-standard (exotic) smooth structure.

References [1] A. Hatcher, “Vector Bundles and KTheory”, 2003. http://pi.math.cornell. edu/~hatcher. [2] M.A. Kervaire, J.W. Milnor, “Groups of homotopy spheres. I”, Ann. of Math. (2) 77 (1963), 504–537.

[5] J.W. Milnor, J.D. Stasheff, “Characteristic classes”, Annals of Mathematics Studies 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.

[3] A. Kupers, “Oriented cobordism: calculation and application”, Harvard University (2017).

[6] G. Reeb, “Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique”, C. R. Acad. Sci. Paris, 222 (1946), 847–849.

[4] J. Milnor, “On manifolds homeomorphic to the 7-sphere”, Ann. of Math. (2) 64 (1956), 399– 405.

[7] G. Tiozzo, “Differentiable structures on the 7sphere”, University of Toronto (2009).

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AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Random walks on supersingular isogeny graphs ∗ Enric

Florit Zacarı́as

Universitat de Barcelona. efz1005@gmail.com ∗Corresponding author

Resum (CAT) Aquest article dona una visió general de les corbes el·lı́ptiques supersingulars i dels seus grafs d’isogènies. Els grafs d’isogènies han guanyat atenció durant els darrers quinze anys gràcies a les seves aplicacions per construir protocols criptogràfics resistents a atacs quàntics. El seu estudi involucra parlar de corbes el·lı́ptiques, d’àlgebres de quaternions i de passeigs aleatoris sobre grafs (quasi) regulars. En aquest text, donem les eines necessàries per establir la propietat de Ramanujan, que connecta corbes supersingulars en caracterı́stica p amb formes modulars de nivell p. A mode d’aplicació, expliquem la funció de hash de Charles, Lauter i Goren.

Abstract (ENG) We survey several aspects of supersingular elliptic curves and their isogeny graphs. Isogeny graphs have obtained attention for the last fifteen years due to their uses in quantum-resistant cryptographic protocols. Studying them involves looking at elliptic curves, quaternion algebras, and random walks on (almost) regular graphs, among other topics. In particular, we give the tools necessary to state the Ramanujan property, connecting supersingular curves in characteristic p with modular forms of level p. We also explain the hash function of Charles, Lauter and Goren as an example of application.

Keywords: isogenies, Ramanujan graphs, random walks. MSC (2020): Primary 14H52, 05C48. Secondary 11F11. Received: July 31, 2021. Accepted: September 22, 2021.

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Acknowledgement The author has been partially supported by a Màster+UB grant at the IMUB.

Reports@SCM 6 (2021), 23–34; DOI:10.2436/20.2002.02.25.

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1. Introduction The main goal of this article is to give an overview of the following result. Theorem 1.1. All isogeny graphs of supersingular elliptic curves are Ramanujan. Supersingular elliptic curves have received an increasing amount of attention due to their applications in postquantum cryptography. While classical elliptic curve cryptography relied on the group structure of elliptic curves over finite fields, isogeny-based cryptography uses morphisms between curves to work with large, random-looking graphs. The Ramanujan property is often cited when introducing supersingular cryptographic protocols, although it is usually not fully explained. This article is an attempt to present the main ingredients that go into this property. The structure of isogeny graphs has been studied at large by Hecke, Eichler, Pizer, Mestre and Kohel, among others (see e.g. [11, 12, 8, 9]). Here we concentrate on the supersingular case, which in turn requires looking at quaternion algebras and modular forms. The combinatorial properties of Ramanujan graphs are stated more naturally in terms of random walks on Markov chains, so we also use that language. To illustrate the extent to which the Ramanujan property applies to cryptography, we give the construction of the hash function of Charles, Lauter and Goren [2]. The CLG hash function can be applied to any graph that is a good expander, that is, such that any random walk reaches any vertex with uniform distribution fairly quickly. Ramanujan graphs such as supersingular isogeny graphs are optimal expanders. The text is organised as follows. We first review the necessary facts about elliptic curves, a complete treatment is found in [13, Chs. II, III and V]. We then define isogeny graphs and their random walks. Afterwards we give background on modular forms and Hecke operators – the reader is invited to consult [3] and [4, Chs. 1 and 5] for further details. We then sketch the proof of the Ramanujan property. The final section is dedicated to the CLG hash function.

2. Elliptic curves Definition 2.1. Let K be a field. An elliptic curve E/K is a genus one smooth projective curve defined over K together with a K -rational point, denoted by ∞ ∈ E (K ). By the Riemann–Roch theorem [13, Th. II.5.4, Prop. III.3.1], any elliptic curve E/K has an affine Weierstrass equation E : y 2 + a1 xy + a3 y = x 3 + a2 x 2 + a4 x + a6 . The corresponding K -rational point appears as the unique point at infinity ∞ = [0 : 1 : 0]. If char K is different from 2 and 3, a change of variables puts E in short Weierstrass form, E : y 2 = x 3 + Ax + B with A, B ∈ K . The smoothness condition amounts to the discriminant ∆E = 4A3 + 27B 2 being different from zero. Given a field L containing K , one can define an abelian group structure on E (L), given by algebraic morphisms and such that ∞ is the identity. This makes E a one-dimensional instance of an abelian variety.

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An important quantity associated to E is its j-invariant, j(E ) = 1728

4A3 . 4A3 + 27B 2

Two elliptic curves are isomorphic over K̄ (as algebraic varieties and as groups) if and only if their j-invariants are equal. Sometimes special care has to be taken, since the isomorphism could be defined over a (finite) extension of the base field. Definition 2.2. A non-constant morphism of curves φ : E1 → E2 mapping ∞E1 to ∞E2 is called an isogeny. The degree of φ, deg φ, is defined to be the degree of the extension of fields K (E1 )/φ∗ K (E2 ). The isogeny φ is called separable or (purely) inseparable according to the nature of this extension. The theory of curves implies that any isogeny φ : E1 → E2 has to be surjective. Moreover, being a map of abelian varieties, φ induces a group homomorphism E1 (K̄ ) → E2 (K̄ ). Since the dimensions of E1 and E2 are equal, the kernel of φ is a finite group. One has deg φ = # ker φ if and only if φ is a separable isogeny. Conversely, we have the following result. Theorem 2.3 ([13, Thm. III.4.12]). Let E be an elliptic curve over K , and let G be a finite subgroup of E . Then, there exist an elliptic curve E 0 and a separable isogeny φ : E → E 0 , both unique up to isomorphism, such that ker φ = G . Moreover, φ is defined over the smallest extension L of K such that G σ ⊂ G for every σ ∈ Gal(K̄ /L). This theorem is made effective by the formulas of Vélu [14] which compute φ in time and space linear √ in the size of G . A recent development [1] lowers the complexity to O( #G ) operations and yields an actual speedup when #G is large enough. When K is a finite field Fq for some prime power q = p r , the group E (F̄q ) consists entirely of torsion points. On the other hand, End(E ), the ring of isogenies φ : E → E with addition and composition, is always a ring of characteristic zero. This is due to the injection Z ,→ End(E ) given by mapping an integer n to the multiplication-by-n map, [n] : E → E |n|

P 7→ P + · · · + P. Proposition 2.4 ([13, Cor. 6.4]). Let E be an elliptic curve and let n be a nonzero integer. (i) deg[n] = n2 . (ii) If char K = 0 or char K = p > 0 with p - n, then E [n] := ker[n] ∼ = (Z/nZ)2 . (iii) If char K = p > 0, then either E [p r ] = {0} or E [p r ] ∼ = Z/p r Z for all r ≥ 1.

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Therefore [p] ∈ End(E/Fp ) is the first example of an inseparable isogeny, while for p - n, [n] is the unique separable isogeny with kernel E [n] given by Theorem 2.3. If G ⊂ E is a finite cyclic subgroup of order n, and φ : E → E 0 = E /G is its quotient isogeny, then the isogeny ψ : E 0 → E 0 /φ(E [n]) satisfies that ψ ◦ φ has kernel E [n], so E 0 /φ(E [n]) ∼ = E and ψ ◦ φ ∼ = [n]. This motivates the following definition-result. Proposition 2.5 ([13, Thm. 6.1]). Let φ : E1 → E2 be an isogeny. Then there exists an isogeny φ̂ : E2 → E1 called the dual isogeny, such that φ ◦ φ̂ = φ̂ ◦ φ = [deg φ]. (p)

Example 2.6. Given a curve E/Fq : y 2 = x 3 + Ax + B, there is a curve E/Fq : y 2 = x 3 + Ap x + B p . The curve E and its p-power E (p) are related by the Frobenius isogeny π : (x, y ) 7→ (x p , y p ). This is a purely inseparable isogeny, so its kernel is trivial. In fact, all inseparable isogenies factor as a power of π composed with a separable isogeny. We have deg π = p, so that π̂ ◦ π = [p]. It is important to establish the group of automorphisms of E , namely, the group of isogenies of degree 1. This is always a finite group, which we denote by Aut(E ). Proposition 2.7 ([13, §III.10]). Let E be an elliptic curve defined over K , with either char K = 0 or char K = p ≥ 5. (i) If j(E ) = 0, then Aut(E ) ∼ = Z/6Z. (ii) If j(E ) = 1728, then Aut(E ) ∼ = Z/4Z. (iii) Otherwise, Aut(E ) = h[−1]i ∼ = Z/2Z. Let B be a finite-dimensional algebra over Q. We say a subring of√B is an order if its rank as a Z-module is finite and equal to the dimension of B. In a quadratic field Q( D) there is a unique maximal order. Meanwhile, in a quaternion algebra B = Q + iQ + jQ + ijQ,

i 2 , j 2 ∈ Q× ,

ij = −ji,

there are multiple maximal orders. The endomorphism ring of an elliptic curve defined over a finite field can be classified as follows [13, Thm. V.3.1]. Theorem 2.8. Let E/Fq be an elliptic curve. The endomorphism ring of E is either (i) An order in an imaginary quadratic field, or (ii) An order in the unique definite quaternion algebra over Q ramified at p and ∞. We call (i) the ordinary case, while (ii) is called the supersingular case. The supersingular case can be characterized in several ways [13, Thm. V.3.1].

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r

r

Theorem 2.9. Let E/Fq be an elliptic curve. For each integer r ≥ 1, let πr : E → E (p ) , π̂r : E (p ) → E be the p r -power Frobenius and its dual. The following are equivalent: (i) E [p r ] = 0 for one (all) r ≥ 1. (ii) π̂r is purely inseparable for one (all) r ≥ 1. (iii) The map [p] : E → E is purely inseparable and j(E ) ∈ Fp2 . (iv) End(E ) is an order in a quaternion algebra. The third condition implies that there is a finite number of isomorphism classes of supersingular elliptic curves. It is possible to give the actual number of classes [13, §V.5]. p Theorem 2.10. There are 12 + εp isomorphism classes of supersingular elliptic curves over F̄p , where εp = 0, 1, 1 or 2 depending on whether p ≡ 1, 5, 7 or 11 mod 12. Lemma 2.11 ([13, Exs. V.4.4, V.4.5]). (i) The curve E/Fp : y 2 = x 3 + x (which has j(E ) = 1728) is supersingular if and only if p ≡ 3 mod 4. (ii) The curve E/Fp : y 2 = x 3 + 1 (which has j(E ) = 0) is supersingular if and only if p ≡ 2 mod 3.

2.1 The Deuring correspondence Let E be a supersingular elliptic curve defined over a finite field Fq . Let O = End(E ) and B = O ⊗ Q = End0 (E ). We fix a “base” curve E0 with its corresponding endomorphism ring O0 . The Deuring correspondence is the following functorial equivalence: Invertible left O0 -modules, with Supersingular elliptic curves . ⇐⇒ nonzero left O0 -module homomorphisms over Fq with isogenies We follow [15, Ch. 12]. Given a nonzero left ideal I ⊂ O, we want to assign it a curve EI and an isogeny φI : E → EI . If I contains a separable isogeny α : E → E , then we can define E [I ] = {P ∈ E | P ∈ ker β ∀β ∈ I }, and assign the isogeny φI : E → EI = E /E [I ]. If all isogenies in I are separable, we can write I = P r I 0 , where P = (π) is the ideal generated by the p-Frobenius, and I 0 contains a separable r r r isogeny. We then have the isogeny φI : E → E (p ) → EI , where EI = E (p ) /E (p ) [I 0 ]. The correspondence also works in the opposite way: given an isogeny φ : E → E 0 , there exists a left ideal I of O and an isomorphism ρ : EI → E 0 such that φ = ρφI . If we let I be a left ideal for O0 = End(E0 ), and consider Hom(EI , E0 ) (which is a left End(E0 )-ideal), then the morphism Hom(EI , E0 ) → I given by ψ 7→ ψφI is an isomorphism of left O0 -ideals. The full correspondence is summarized by the following result [15, Th. 42.3.2]. Theorem 2.12. The association E 7→ Hom(E , E0 ) is functorial and defines an equivalence between the category of supersingular elliptic curves over Fq with isogenies, and the category of invertible left O0 -modules with left O0 -module homomorphisms. Further details on quaternion ideals in relation to isogenies can be found in Voight’s book and in the articles by Kohel [8, 9].

Reports@SCM 6 (2021), 23–34; DOI:10.2436/20.2002.02.25.

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3. Isogeny graphs Definition 3.1. Let p ≥ 5 and ` be two different primes. The supersingular `-isogeny graph, denoted Γ(`; p), is the directed multigraph defined as follows: • The vertices are isomorphism classes of supersingular elliptic curves defined over F̄p . If E is such a curve, [E ] denotes its corresponding vertex. • The edges [E1 ] → [E2 ] are given by (separable) isogenies E1 → E2 of degree `. These are identified in the following way: fixing the curve E1 , two isogenies φ : E1 → E2 , ψ : E1 → E3 are equivalent if ker φ = ker ψ (and so E2 ∼ = E3 ). We consider all our isogenies to be defined over F̄p . The degree of the vertices of this graph is given by the following fact about finite abelian groups. Lemma 3.2. Let ` be a prime. The group Z/`Z × Z/`Z has ` + 1 cyclic subgroups of order `. The definition says Γ(`; p) is a directed graph, and by the lemma, it has regular out-degree equal to `+1. However, Proposition 2.7 and Lemma 2.11 tell us that all vertices in Γ(`; p) have the same automorphism group whenever p ≡ 1 mod 12. This is the only phenomenon that can make the graph non-undirected. Indeed, if we take two elliptic curves E1 and E2 , and we let w (E1 , E2 ) and w (E2 , E1 ) be the number of `-isogenies from E1 to E2 and viceversa, we have # Aut(E1 ) w (E2 , E1 ) = . # Aut(E2 ) w (E1 , E2 )

(1)

Hence Γ(`; p) is an (` + 1)-regular undirected graph if and only if p ≡ 1 mod 12. To go deeper into the combinatorial structure of isogeny graphs we need to study random walks. Let G = (V , E ) be a directed graph, with a finite set of vertices V and a finite collection of directed edges E . We allow for loops and multiple edges, that is, given two (possibly equal) vertices u and v , there can be two different edges e1 , e2 ∈ E from u to v . We assume that for each edge u → v there is at least one edge v → u, and such that we can reach the whole graph from any single vertex (in particular, G is strongly connected). Given a vertex u ∈ V , we define deg u to be the number of edges coming out of u. If v is another vertex, we let w (u, v ) be the number of edges from u to v . Definition 3.3. Given any u0 ∈ V , a random walk on G is a sequence of vertices u0 , u1 , ... , un , ... , such that w (u, v ) P(ui+1 = v | ui = u) = . deg u This means that, at each step in the walk, we choose our next step to be a vertex neighboring our current position in the graph. Since the state at any given time determines the probabilities for the next state, this is an example of a discrete-time Markov chain. We want to give a probability distribution φ on V that is “stationary” with respect to the random walk. This will mean that, if we sample a vertex u0 according to φ and we take a random walk starting at u0 , we will obtain at each step a vertex that will also be sampled according to φ. Moreover, we will have convergence to φ independently of our starting distribution. We first need an additional hypothesis on G .

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Definition 3.4. Let u ∈ V be any vertex. The period of u is the greatest common divisor of the lengths of all paths on G starting and ending at u. The period of G is the greatest common divisor of the periods of all vertices. We say G is aperiodic if its period is equal to 1. Let us assume that G is aperiodic, undirected, and regular of degree d. If we let M be the adjacency matrix of M, then each column of M sums to d. More importantly, if we let W = d1 M, and we let ψ ∈ R|V | be a vector with positive entries adding to 1, then the multiplication W ψ yields another vector with positive entries adding to 1. The matrix W representing the random walk is called a stochastic matrix. This procedure is equivalent to performing a step in the random walk of G after sampling a vertex according with the distribution on V given by ψ. This motivates the result that follows, whose proof is found in [7, Sec. 3]. The matrix W is symmetric, so its eigenvalues are real and can be ordered as 1 = λ1 > λ2 ≥ · · · ≥ λn > −1. We let λ? = max{|λ2 |, |λn |}. Proposition 3.5. Let G = (V , E ) be a connected, aperiodic undirected graph (possibly with loops and multiple edges). Let u0 ∈ V , and let u0 , u1 , ... , un , ... be a random walk as defined above. Then, for all positive integers n and all vertices v ∈ V we have

1

≤ λn? .

P(un = v ) −

|V |

In particular, φ(u) = 1/|V | is the stationary distribution of the random walk on G . We now consider a family {Gn }n of d-regular graphs with growing number of vertices. A theorem of Alon and Boppana ([7, Th. 2.7]) says that √ 2 d −1 . lim inf λ? (Gn ) ≥ n d √ Definition 3.6. A graph G is said to be Ramanujan if λ? (G ) ≤ 2 d − 1/d. When G is not a regular graph, a similar result can be proven, with stationary distribution given by φ(u) = deg u/(2|V |). Directed graphs require a more careful treatment (even with our strong connectivity hypothesis), as they are very close to general Markov chains. Luckily, the isogeny graphs Γ(`; p) satisfy the nice symmetry property of equation (1). This allows us to give the next result, which generalizes well to abelian varieties (see [5] for further details). Theorem 3.7. Consider the graph Γ(`; p), a starting curve E0 , and a random walk E0 , E1 , ... , En , ... P elliptic The stationary distribution is given by φ = φ̃/ E φ̃(E ) , where φ̃(E ) = 1/# Aut(E ). Given a positive integer n and any supersingular elliptic curve E , s # Aut(E ) . |P(En ∼ = E ) − φ(E )| ≤ λn? # Aut(E0 ) In the particular case that p ≡ 1 mod 12 we recover Proposition 3.5.

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Proof. The stationary distribution equals φ because of the equation (1). For the convergence statement, see [10, Th. 12.3] or [5, Th. 1]. An interesting related result is [6, Th. 1], which states the convergence to the same distribution for the graph Γ({`1 , ... , `r }; p) of isogenies of several degrees. We remark that we still have not shown that Γ(`; p) is a connected graph. The theory of random walks still works, which is why we have omitted this consideration: one can simply apply the results to each connected component of the graph. However, one can indeed show that Γ(`; p) is connected. This will be √ an immediate consequence of the Ramanujan property: if λ? (Γ(`; p)) is bounded by 2 `/(` + 1), then the random walk matrix has a unique eigenvalue equal to 1. A standard fact in algebraic graph theory then says that Γ(`; p) has to be connected. Likewise, the aperiodic property of the graphs also follows from the Ramanujan property: another bit of algebraic graph theory says that a graph is non-bipartite if and only if λ? (Γ(`; p)) < 1. For a strongly connected graph, being aperiodic is equivalent to being non-bipartite.

4. Modular forms and the Ramanujan–Petersson conjecture Let N be a positive integer. We consider the group of integer matrices

a b

Γ0 (N) = a, b, c, d ∈ Z, N | c, ad − bc = 1 . c d

In words, Γ0 (N) consists of integer matrices with determinant 1 whose reduction modulo N is upper triangular. It is a finite index subgroup of SL2 (Z), the group of 2-by-2 integer matrices with determinant 1. These matrices act on the complex upper-half plane H = {τ ∈ C | Im(τ ) > 0} by means of the Möbius transformation, aτ + b a b τ 7→ ·τ = . c d cτ + d Definition 4.1. Let f : H → C be a holomorphic function, and let N and k be positive integers. We say f is a modular form of weight k and level N if a b (i) For all matrices γ = ∈ Γ0 (N), f (γτ ) = (cτ + d)k f (τ ), and c d (ii) For all matrices γ ∈ SL2 (Z), the transformed function f (γτ )(cτ +d)−k is bounded when Im(τ ) → ∞. We denote by Mk (Γ0 (N)) the C-vector space of modular forms of weight k with respect to Γ0 (N). Modular forms can seem somewhat complicated after giving this definition. However, a prominent feature of the theory is that we can work with them using Fourier expansions. For all levels N ≥ 1, the 1 1 matrix T = is in Γ0 (N), the symmetry of f ∈ Mk (Γ0 (N)) implies f (τ + 1) = f (τ ) for all z ∈ H. 0 1 From the fact that f is holomorphic and bounded at infinity, we have a Fourier expansion of the form X f (τ ) = an (f )e 2πinτ . n≥0

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P By letting q = e 2πiτ , we can rewrite this as the q-expansion f (q) = n≥0 an (f )q n . If the Fourier expansion of f (γτ )(cτ + d)−k starts with a0 = 0 for all γ ∈ SL2 (Z), we say f is a cusp form. The subspace of cusp forms is denoted by Sk (Γ0 (N)). Both Mk (Γ0 (N)) and Sk (Γ0 (N)) have finite dimension; see [4, Thms. 3.5.1 and 3.6.1]. We need a few facts about the case of level 1. When the weight k is odd, since the matrix − Id2 is in SL2 (Z) we have f (τ ) = (−1)k f (τ ) for all τ ∈ H, so there are no nonzero modular forms of odd weight. When k ≥ 4 is even, an example of level 1 form is given by the Eisenstein series X 1 Gk (τ ) = (mτ + n)k 2 (m,n)∈Z \{(0,0)}

which is seen to converge uniformly on compact sets. Showing the invariance under SL2 (Z) is then a simple exercise, done by reordering of the series. This can however not be done in weight 2. In fact, we have M2 (Γ0 (1)) = {0}. A proof can be found in [3, Thm. 5.3]. From now on we restrict to the case of weight k = 2. To help study the structure of the spaces M2 (Γ0 (N)), one defines a series of linear operators T` : M2 (Γ0 (N)) → M2 (Γ0 (N)) for each prime `, called the Hecke operators. These operators preserve the subspace of cusp forms. They commute pairwise, and for all ` - N, they are self-adjoint with respect to a certain inner product – called the Petersson scalar product – which is defined as follows: given two cusp forms f and g of level 2, Z hf , g i = f (τ )g (τ ) dx dy . H/Γ0 (N)

By the spectral theorem, the space S2 (Γ0 (N)) has a basis of simultaneous eigenvectors (called P eigenforms) for all Hecke operators T` with ` - N. The effect of each T` on the q-expansion of f (q) = n an q n is given by the formulas (P P n/` + ` an q `n , if ` - N, `|n an q (2) T` (f ) = P n/` , if ` | N. `|n an q P From equation (2), if f (q) = n≥1 an (f )q n is a simultaneous Hecke eigenform such that a1 (f ) = 1 (this is called a normalized eigenform), then we have T` (f ) = a` (f )f , so that the Fourier coefficient a` (f ) is the eigenvalue of f at `. At this point, one usually defines a subspace of S2 (Γ0 (N)) consisting of all forms coming from levels M dividing N: the space of old forms S2old (Γ0 (N)). The space of new forms S2new is then defined as the orthogonal of S2old with respect to h·, ·i. Since we are only interested in weight 2 and prime level p, and there are no forms of weight 2 and level 1, we have S2 (Γ0 (p)) = S2new (Γ0 (p)). It can be seen that this space has a basis of simultaneous eigenforms for all Hecke operators. What we shall do below is to relate the space of cusp forms of level p, together with its structure as a Hecke module, with the adjacency matrices of the graphs {Γ(`; p)}` . To that effect, we need the following bound on the Hecke eigenvalues. Theorem 4.2 (Ramanujan–Petersson conjecture, proven by Deligne). Let f ∈ S2new (Γ0 (N)) be a normalized Hecke eigenform with Fourier expansion X f (q) = an (f )q n . n>0

√ Then for any prime ` we have the inequality |a` (f )| ≤ 2 `.

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5. Isogeny graphs are Ramanujan L Let p be a prime. We define a formal group Mp = E Z[E ], where E ranges over the set of isomorphism classes of supersingular elliptic curves over F̄p . We let αE = # Aut(E )/2. The space Mp has a scalar product defined by hE , E i = αE , hE , E 0 i = 0 for E ∼ 6 E 0 , and extended linearly. = We give Mp a structure of Hecke module as follows. For each prime ` 6= p, we define the `th Hecke operator by X [E /C` ], T` [E ] = C`

where C` runs over all order-` cyclic subgroups of E , and E /C` is the image of the unique separable isogeny with kernel C` . These operators commute for `1 , `2 6= p. Other operators which we do not need can be defined [11]. The connection of Mp with our isogeny graphs is clear: the adjacency matrix of Γ(`; p) is the matrix of the operator T` in the basis {[E ]}E of Mp . P 1 We let Eis = αE [E ] ∈ Mp ⊗ Q and let

X nX o

Mp0 = xE [E ]

xE = 0 be the subspace orthogonal to Eis. We have T` Eis = (` + 1) Eis for all primes `. Theorem 5.1. There is an isomorphism of Hecke modules Mp0 ⊗ C ∼ = S2 (Γ0 (p)). Sketch of proof. The isomorphism is constructed in two steps. The first one involves the Deuring correspondence. After fixing a base curve E0 , the functor E 7→ Hom(E , E0 ) takes a supersingular elliptic curve over L F̄p to a left End(E0 )-ideal. One then builds an analog module of left ideals for the order End(E0 ), N = I Z · [I ]. This module has an inner product h·, ·i and an action of Hecke operators X [J]. T` ([I ]) = φ : I →J J/φ(I )∼ =(Z/`Z)2

Now there is a Hecke-bilinear pairing Θ : N × N → M2 (Γ0 (N)). To define it, we first define a collection of operators An on Mp in terms of the operators T` , and such that A` = T` for all primes `. Then Θ is defined by ∞ X Θ([I ], [J]) = 1 + 2 hAn ([I ]), [J]iq n n=1

and extended bilinearly to N × N. This pairing takes N × N 0 and N 0 × N to the space of cusp forms. For all ` 6= p one has T` (Θ([I ], [J])) = Θ(T` ([I ]), [J]) = Θ([I ], T` ([J])) and then the proof is concluded by showing that Θ(−, v ) : N → M2 (Γ0 (p)) is an isomorphism of Hecke modules for an appropriate v ∈ N (see [8, Cor. 4]). √ Corollary 5.2. The second largest eigenvalue of the graph Γ(`; p) is bounded by 2 `. In particular, for p ≡ 1 mod 12 the supersingular isogeny graph Γ(`; p) is an (` + 1)-regular Ramanujan graph.

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Proof. The matrix of the Hecke operator T` on Mp is the adjacency matrix of Γ(`; p). It has largest eigenvalue λ1 = ` + 1 corresponding to the constant vector 1. The remaining eigenvalues satisfy the Ramanujan–Petersson bound for normalized √eigenforms in S2 (Γ0 (N)), so that the second largest eigenvalue of the random walk matrix satisfies λ? ≤ 2 `/(` + 1). This is the definition of Ramanujan graph whenever the graph is regular and undirected, which is the case whenever p ≡ 1 mod 12. Together with Theorem 3.7, this gives a bound on the diameter of Γ(`; p) which is logarithmic in p. This has the following arithmetic formulation. Theorem 5.3. Let E1 and E2 be two supersingular elliptic curves over F̄p . For any prime ` 6= p, there exists a separable isogeny φ : E1 → E2 of degree `O(log p) .

6. Hash functions from isogeny graphs F We fix a prime number `. Consider the set {0, 1, ... , ` − 1}∗ = n≥1 {0, 1, ... , ` − 1}n of strings of arbitrary length, representing information codified in base `. For example, the case ` = 2 corresponds to arbitrarylength binary strings. Given a finite set S, a hash function consists of a map h : {0, ... , ` − 1}∗ → S. A cryptographic hash function usually requires additional properties, for instance, it asks that the image of h is “uniformly distributed” in some sense. Using Theorem 3.7 and the fact that Γ(`; p) is a Ramanujan graph, we shall build a hash function with the property that for a fixed m large enough, the image of h : {0, 1, ... , ` − 1}m → S is uniformly distributed. We follow the construction of Charles, Lauter and Goren [2]. The set S shall consist of the isomorphism classes of supersingular elliptic curves. For the setup, we pick two such elliptic curves E−1 , E0 which are connected by an isogeny of degree `, φ−1 : E−1 → E0 . Then there are ` isogenies of degree ` from E0 which (0) (`−1) are different from the dual φ̂−1 . We label them φ0 , ... , φ0 . Given a string b = b0 · · · bm−1 ∈ {0, 1, ... , ` − 1}m , we define recursively a path in Γ(`; p) by taking a step in the graph according to bi . The path starts by (b ) φ0 0 : E0 → E1 . Recursively, after taking an isogeny Ei−1 → Ei we label the ` isogenies from Ei different (0) (`−1) (b ) from φ̂i−1 as φi , ... , φi , and take a further step by using φi i . After m steps, we will arrive at a supersingular elliptic curve Em . The hash of b can then be defined to be this curve (in the set of supersingular elliptic curves, which achieves a uniform distribution after sufficiently many steps). Usually, one takes the j-invariant of the curve, as it depends only on its isomorphism class. The CLG hash function has two additional properties: collision resistance and preimage resistance. These are due to the lack of short loops in Γ(`; p) for suitable p. Additional details can be found in [2].

References [1] D.J. Bernstein, L. De Feo, A. Leroux, B. Smith, Faster computation of isogenies of large prime

degree, in: Proceedings of the Fourteenth Algorithmic Number Theory Symposium, The Open

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Book Series 4, Mathematical Science Publishers, Berkeley, CA, 2020, 39–55. [2] D.X. Charles, K.E. Lauter, E.Z. Goren, “Cryptographic hash functions from expander graphs”, J. Cryptology 22(1) (2009), 93–113.

[9] D. R. Kohel, Hecke module structure of quaternions, in: Class field theory—its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo, 2001, 177–195.

[3] K. Conrad, “Modular forms (draft, CTNT [10] D.A. Levin, Y. Peres, E.L. Wilmer, Markov chains and mixing times, With a chapter by 2016)”, https://ctnt-summer.math.uconn James G. Propp and David B. Wilson, American .edu/wp-content/uploads/sites/1632/ Mathematical Society, Providence, RI, 2009. 2016/02/CTNTmodularforms.pdf. [4] F. Diamond, J. Shurman, A first course in [11] J.-F. Mestre, La méthode des graphes. Exemmodular forms, Graduate Texts in Mathematples et applications, in: Proceedings of the inics 228, Springer-Verlag, New York, 2005. ternational conference on class numbers and fundamental units of algebraic number fields [5] E. Florit, B. Smith, “Automorphisms and (Katata, 1986), Nagoya Univ., Nagoya, 1986, isogeny graphs of abelian varieties, with ap217–242. plications to the superspecial Richelot isogeny graph”, Preprint (2021), https://arxiv. [12] A.K. Pizer, “Ramanujan graphs and Hecke oporg/abs/2101.00919. erators”, Bull. Amer. Math. Soc. (N.S.) 23(1) [6] S.D. Galbraith, C. Petit, J. Silva, “Identifica(1990), 127–137. tion protocols and signature schemes based on supersingular isogeny problems”, J. Cryptology [13] J.H. Silverman, The arithmetic of elliptic 33(1) (2020), 130–175. curves, Second edition, Graduate Texts in Mathematics 106, Springer, Dordrecht, 2009. [7] S. Hoory, N. Linial, A. Wigderson, “Expander graphs and their applications”, Bull. Amer. [14] J. Vélu, “Isogénies entre courbes elliptiques”, Math. Soc. (N.S.) 43(4) (2006), 439–561. C. R. Acad. Sci. Paris Sér. A-B 273 (1971), [8] D.R. Kohel, Computing modular curves via A238–A241. quaternions, in: Fourth CANT Conference: Number Theory and Cryptography, University [15] J. Voight, Quaternion algebras, Graduate Texts in Mathematics 288, Springer, Cham, 2021. of Sydney, Dec. 3, 1997.

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AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Self-similar profiles in Analysis of Fluids. A 1D model and the compressible Euler equations ˚ Gonzalo

Cao-Labora

Massachusetts Institute of Technology (MIT). gcaol@mit.edu ˚Corresponding author

Resum (CAT) Presentem dos nous resultats en anàlisi de fluids relacionats amb l’existència de singularitats fent servir perfils autosimilars i anàlisi d’estabilitat al voltant d’ells. El primer resultat és una nova prova de la formació de singularitats per l’equació d’Okamoto–Sakajo–Wunsch amb petit paràmetre fent ús d’un perfil autosimilar aproximat. A la segona part trobem nous perfils autosimilars, radials i suaus, per a l’equació d’Euler compressible i isentròpica. Aquest és el primer perfil d’aquest tipus trobat pel cas de gasos monoatòmics.

Abstract (ENG) We present two new results in Analysis of Fluids involving the existence of singularities via self-similar profiles and stability analysis around them. The first result is a new proof of the formation of singularities for the Okamoto–Sakajo–Wunsch equation with small parameter, which is done via a stability analysis around an approximate self-similar profile.The second result consists on the finding of new smooth radial self-similar profiles developing singularities for the isentropic compressible Euler equations. This is the first proof of such profile for the monatomic gas case.

Keywords: compressible Euler, Okamoto–Sakajo–Wunsch, self-similar profiles, modulation variables. MSC (2010): 35Q35, 76B03, 35L65, 76N10. Received: July 30, 2021. Accepted: November 25, 2021.

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Acknowledgement G. C.-L. was supported by the CFIS mobility program during the development of this work and also received partial support by the MOBINT program.

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1. Introduction The Navier–Stokes equation in three dimensions models the behaviour of a non-compressible fluid and is given by Bt u ` u ¨ ∇u “ ´∇p ` ν∆u, (1) where x P R3 is physical space, the velocity of the fluid is given by the vector upx, tq “ pu1 , u2 , u3 q, we denote the viscosity by ν and ppx, tq is the pressure. We will always denote with u ¨ ∇u the vector whose i-th component is u ¨ ∇ui . The previous equation can be interpreted in terms of Newton’s second law, because the left hand-side is the convective derivative of the velocity (derivative along the trajectories of the fluid), and the right-hand side is the force exerted on the fluid. In order to close (1) (which has 4 unknowns p and ui but only a 3-component equation), one notes that the incompressibility of the fluid implies divpuq “ 0 (volumes are preserved along trajectories), which together with (1) forms a closed system. In order to drop the pressure one can take the curl in (1) and define the vorticity to be ω “ curlpuq, obtaining pBt ` u ¨ ∇qω “ ω ¨ ∇u ` ν∆ω. (2) The incompressibility condition div u “ 0 ensures that u can be recovered from ω via a nonlocal operator, thanks to the Biot–Savart law: ż 1 px ´ y q ^ ωpy q upxq “ p.v. , (3) 4π |x ´ y |3 R3 where p.v. refers to the fact that the integral is done in the principal value sense1 . The system formed by (2) and (3) has been widely studied, and the smooth existence of solutions (or a counterexample) would solve the famously known Millenium Clay Problem [14]2 . A principal difficulty of this problem is the fact that there is a nonlocal operator in the RHS, due to the fact that u is recovered from ω via a nonlocal operator. In order to reflect this quadratic nonlinear term, Constantin, Lax and Majda [8], introduced the following one dimensional model equation: ωt “ ´2ωHω.

(4)

Here H represents the Hilbert Transform which is a nonlocal operator in dimension 1, given by Hω “ that it is bounded as H : L2 pRq Ñ for all f P H 1 pRq3 . A reference introducing the Hilbert Transform is [25].

ş ωpy q 1 π p.v. x´y dy . Some important properties of the Hilbert Transform are L2 pRq, or that it commutes with derivatives, that is Hpf 1 q “ pHf q1

The constant ´2 in (4) makes no special role (it just rescales the solutions), and it is only fixed for the sake of comparing the model with other models. The main advantage of this model is that there are explicit formulas for the solutions, due to the following property of the Hilbert Transform: 1 A principal value integral is done by removing a ball of radius ε around the singularity (in this case y “ x) and taking ε Ñ 0. At infinity, one integrates in a ball Bp0, Rq and takes R Ñ `8 (in the way of a Riemann indefinite integral). 2 To be precise, the Millenium Clay Problem also requires those solutions to be finite energy, that is, up¨, tq uniformly bounded in L2 pRq. 3 For the reader unfamiliar with Sobolev spaces, H 1 pRq is basically the space of f P L2 pRq with a derivative f 1 P L2 pRq.

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Theorem 1.1 (Titchmarsh theorem [26]). Let f , g P L2 pRq and let H Ă C be the upper half-plane of the complex plane. We have that g “ Hf if and only if there exists an holomorphic function on the upper half plane G : H Ñ C such that G px ` iy q converges ş almost everywhere to f pxq ` ig pxq as y Ñ 0. Moreover, in that case, G can be taken in L2 pHq, that is H |G pzq|2 ă 8. From now on, we denote fˇpxq “ f pxq ` iHf pxq, which admits an holomorphic extension to the upper half plane by the previous theorem. Looking at our equation ωt “ ´2ωHω, let G be that holomorphic extension of ω̌. The function iG 2 is holomorphic, and its real part over the real line is ´2ωHω. Therefore, if we solve the ODE Gt “ iG 2 for G p¨, tq holomorphic over the upper half plane, we will recover our G0 pzq solution ω looking at the real part of G over the real line. The solution to that ODE is G pz, tq “ 1´iG , 0 pzqt and the initial data G0 pzq is the holomorphic extension of ω0 ` iHω0 . One can recover ω from the real part of G pz, tq for real z, and obtain the following theorem. Theorem 1.2 (Constantin, Lax and Majda [8]). Let ω be a solution to (4) with ωp¨, 0q “ ω0 P H 1 pRq. Then, we have that ω0 pxq . (5) ωpx, tq “ p1 ` Hω0 pxqtq2 ` ω0 pxq2 t 2 In particular, it develops a singularity if and only if exists some x with ω0 pxq “ 0 and Hω0 pxq ă 0. x Example 1.3. One interesting example of initial data leading to singularity is ω0 pxq “ 1`x 2 , whose Hilbert ´1 x´i 1 Transform is Hω0 pxq “ 1`x 2 because 1`x 2 “ x`i is holomorphic on the upper half plane. As Hω0 pxq “ ´1 ` x ˘ 1 and ω0 pxq “ 0, it develops a singularity. Moreover, equation (5) gives us that ωpx, tq “ 1´t F 1´t , for x F pxq “ 1`x 2 . This is called a self-similar solution, because the solution looks the same for all times t (it looks like F ), just rescaled horizontally and vertically by some time-dependent factor.

2. The Okamoto–Sakajo–Wunsch model The Okamoto–Sakajo–Wunsch model (OSW for short) was introduced in [23] as a a generalization of the CLM model. As one can observe from (4), the CLM model just substitutes the full covariant derivative pBt ` u ¨ ∇qω in 3D Euler with the term ωt , without including a term reflecting u ¨ ∇ω. The OSW model precisely solves that, and as u “ curl´1 pωq in 3D Euler, a reasonable choice in a 1D model is to take ş x upxq “ Λ´1 ω “ ´ 0 ωpy q dy , because it is also an order ´1 nonlocal operator (the ´ sign in front is just a convention). Indeed, the OSW model reads # ωt ` auωx “ ´2ωHω, şx u “ Λ´1 ω “ ´ 0 ω,

(6)

where a is just some fixed parameter. Note that when we write ωx , ωt , we are denoting derivatives with subscripts, we will do so for the rest of this article. The case a “ 0 recovers the CLM model. The cases a “ 2 and a “ ´2 are also interesting, the first one is the De Gregorio model [11] and the second one was introduced by Córdoba, Córdoba and Fontelos, because its solutions yield solutions of the 2D SQG equation with some symmetries [9].

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The question of the existence of singularities for this model remained a long-standing problem. First, Castro and Córdoba showed the existence of solutions developing singularities for a ă 0 [4]. They also showed singularities in finite time for some positive a ą 0, but using non-smooth initial data, so the question remained whether singularities can form with smooth initial data. This was recently solved by Elgindi and Jeong [13, 12], proving singularities can form for odd smooth initial data provided that a ą 0 is small enough. The proof is based on a H 3 pRq stability analysis around the self-similar profile for a “ 0. One does not expect to have explicit formulas for the self-similar profile around some a small enough, but they use y the self-similar profile for CLM, F py q “ 1`y 2 , and this will be very close to the exact self-similar profile provided that a is small enough. The stability analysis consists in showing that the difference between the approximate self-similar profile and the exact solution will remain much smaller than the approximate self-similar profile, so that there is a singularity dominated by this approximate self-similar profile near the blow up time. The difference of our work with the previous work of [13, 12] is that we will do the stability analysis in a space that combines L8 norms for low derivatives with L2 -based norms for higher derivatives (instead of H 3 pRq). In particular, this shows singularities for some non-decaying initial data (since they can be in L8 , but not in L2 ). More importantly, the L8 analysis requires different techniques around the singularity, and we will use the complex structure of the Hilbert Transform, which is at the core of our proof. In this short presentation, we intend to give a brief overview of the proof rather than proving all the estimates. This proof is joint work with Tristan Buckmaster, Javier Gómez-Serrano and Federico Pasqualotto.

2.1 Formulation First of all, let us note that equation (6) gives odd solutions for odd initial data, because the Hilbert Transform changes parity (and therefore Λ´1 respects parity). From now on, when we talk about the OSW model, we will be talking about odd solutions. As we already outlined, the main idea of this strategy is to consider ω solving OSW for a small y enough, substract the self-similar profile F py q “ 1`y 2 and perform some stability analysis for the difference. ` x ˘ 1 F 1´t , but one However, we need to choose the rescaling for our self-similar profile. For CLM we had 1´t 1 expects those factors 1´t to change a bit as we change the parameter a. The fundamental idea to do that temporal rescaling with the so-called modulation variables, which are fixed dynamically. This idea has been used for OSW in [12, 5] and for other equations as 2D Burgers equations with transverse viscosity [7], the Prandtl equation [6, 10], the 2D compressible Euler equation [3, 19] or the 3D compressible Euler equation [2], to cite some. The idea is to consider a rescaled profile ` ˘ 1 of the form λptq F µptqx λptq , where λptq and µptq are called modulation variables, and their evolutions will fixed dynamically in order to satisfy some property4 . In our case, we will use them to fix the scalar quantities ωx p0, tq and Hωp0, tq. We will choose λptq and µptq such that those quantities are completely ` ˘ 1 F µptqx absorbed by the profile, in other words, if we let q “ ω ´ λptq λptq , we will have qx p0, tq “ Hqp0, tq “ 0. Oddness yields qp0, tq “ Hqx p0, tq “ 0 directly, so our way of fixing λ and µ will ensure that q̌ and q̌x vanish at zero, ensuring that the self-similar part is the dominating part close enough to zero. 4

This will agree with the rescaled solutions showed for the CLM model because we will have that both 1 approach 1´t as a Ñ 0.

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As we want to show that our self-similar profile dominates, it will be easier to work in the self-similar variables, where our profile is constant and we only have to worry about the size of the difference. This is µpsqx 1 achieved by fixing ds dt “ λpsptqq (with sp0q “ 0) and y “ λpsq as the new time and space variables. Note that we make λ and µ depend on s instead of t. This will make calculations simpler, since we plan to work in py , sq variables. If we write (6) in py , sq, the equation for q reads µs qs ` y pFy ` qy q ´ µ

ˆ

˙ λs ` 1 pF ` q ` yFy ` yqy q “ Mq ` N ` aJ̃. λ

(7)

Here, N is a quadratic term including all the terms of the form qHq and qy Λ´1 q. It will not be relevant because we will work with }q}X ď ε, so }N}X À ε2 is much smaller (X is an appropriate Banach space that we will introduce later). The term aJ̃ is an inhomogeneous term independent of q (only depending on F ) and it will be small because a is small enough. The term Mq is a linear term in q and will be the important term that does not allow q to grow a lot, yielding stability.

2.2 The modulation In order to control (7), we need to control the modulation variables µpsq and λpsq. As we already told, we are going to fix λ and µ such that q 1 p0q “ Hqp0q “ 0, which gives us a second order cancellation of q̌ around the origin. Therefore, we will assume that the initial data satisfy q01 p0q “ 0, Hq0 p0q “ 0 and we will get a system of evolution equations for pλ, µq just by taking a spatial derivative or a Hilbert Transform d d in (9) and asking for ds qy p0q “ 0 and ds Hqp0q “ 0. Doing so, we get $ˆ ˙ λs ’ ´1 ´1 ´1 ’ ’ & λ ` 1 ´ ak2 “ aHpFy Λ q ` qy Λ F ` qy Λ qqp0q, ˙ ˆ ’ µ s ’ ’ ´ ak1 “ 2aHpFy Λ´1 q ` qy Λ´1 F ` qy Λ´1 qqp0q, % µ

(8)

where k2 “ HpFy Λ´1 F qp0q “ logp2q ´ 1{2 and k1 “ 2k2 ´ 1 “ ´2 ` 2 logp2q. Therefore, in order to control the modulation, we just need to control the quantity HpFy Λ´1 q ` qy Λ´1 F ` qy Λ´1 qqp0q, which seems feasible, because we will take q to be small enough. Using (8), it is useful to rewrite equation (7) as qs “ M0 q ` aM1 q ` aP ` N ` aJ,

(9)

where aP “ pF ` q ´ yFy ´ yqy qHpqy Λ´1 F ` Fy Λ´1 q ` qy Λ´1 qqp0q, M0 q “ ´q ´ yqy ´ 2qHF ´ 2FHq. The term pM0 q ` aM1 qq is the linear term from before, but we have put together all the terms Opaq in aM1 q, and keep the important terms in M0 q. The term aP comes from the modulation we have just discussed. Lastly, N is a non-linear term as before, and aJ is the inhomogeneous term different from the one before. One also has that J̌ and J̌y are zero at the origin.

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2.3 Bootstrap argument The whole argument consists in showing that q remains small in a suitable Banach space under (9). We will assume the initial bounds a }ψq̌0 }L8 ď ε1 {2, }ψByy q̌0 }L8 ď ε2 {2, }y By q0 }L2 ď ε3 {2 and } 1 ` y 2 By4 q0 }L2 ď ε4 {2, (10) for some a ! ε1 ! ε2 ! ε3 ! ε4 ! 1 and ψ “ }q̌p¨, sq}L8 ď ε1 ,

}q̌yy p¨, sq}L8 ď ε2 ,

1`y 2 . y2

Then, we will try to show the uniform bounds

}yqy p¨, sq}L2 ď ε3

a and } 1 ` y 2 By4 qp¨, sq}L2 ď ε4 .

(11)

The core of the proof is the following proposition. Proposition 2.1. Suppose that q0 satisfies (10) and let q be the solution to (9). Then, the bounds (11) hold uniformly in time s. Let } ¨ }X be the norm consisting on adding up all the norms in (10), and the Banach space X formed by functions for which that norm is finite. The norm coming from adding up all the norms in (11) may seem different (because of the ψ weight in L8 ) but is in fact an equivalent norm if we restrict to the space of q with q̌p0q “ 0 and q̌y p0q “ 0. The strategy for proving Proposition 2.1 will be a bootstrap argument, which we are going to describe. The local wellposedness theory (which follows from the general theory of Kato and Lai [18]), gives us that the previous norms evolve continuously with respect to the time s. Therefore, in order to show (11) globally in time, we argue by contradiction, assuming that (11) are satisfied up to some time s0 , and that one of the bounds is broken at time s0 . If we get a contradiction from that, we will have that (11) are global in time. Therefore, for each of the bounds, we just suppose that one is the first to be broken and we arrive to a contradiction. The reasons behind why do we need such variety of norms are mainly the following. First of all, we need to control q̌ and not just q due to the fact that we will work with the complexified equation for q̌ “ q `iHq, as in CLM. Secondly, we also need their second derivatives to get some nontrivial control at the origin (remember q̌p0q “ q̌y p0q “ 0). Also, the L2 norms at higher derivatives avoid a loss of derivative that we face for the estimates on }By2 q̌}8 . Another good property about our norm is that one can derive the inequality }HpFy Λ´1 q ` qy Λ´1 F ` qy Λ´1 qq}L8 ď C }q}X , (12) for some constant C ą 0, which allows to control the modulation.

2.4 Example of a simplified equation Let us illustrate the proof of Proposition 2.1 with a much simpler example. We have that aM1 q and aP will be small because M1 q, P are bounded and a is small enough. On the other hand, N will be small because it is nonlinear, so a term like qHq will be simply bounded by ε21 . Therefore, let us assume we just have to deal with the much simpler equation qs “ M0 q ` aJ, instead of (9). We will show how the bootstrap works for the }q̌}L8 estimates in this equation.

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Lemma 2.2. Let us assume that q satisfies qs “ M0 q ` aJ and the initial data qp¨, 0q “ q0 p¨q satisfies assumptions (10). Then, we have the estimate }q̌p¨, sq}L8 ď ε1 holds for all times s ě 0. } Proof. The point here is that M0 q admits a nice formula for its complexification, namely M 0 q “ ´q̌ ´ 5 s } y q̌ ` 2i F̌ q̌ . Using the characteristic curves y “ ηy0 psq “ y0 e in our equation q̌s “ M q ` a J̌, we obtain 0 that βy0 psq “ q̌pηy0 psq, sq satisfies B βy psq “ ´βy0 ` 2i F̌ pηy0 psqqβy0 psq ` aJ̌pβy0 psqq. Bs 0 The problem with this equation is that 2i F̌ p0q “ 2, which spoils the damping coming from ´βy0 . As F̌ decays, there is no problem for big enough y “ ηy0 psq. To solve the issue near zero we introduce the 2 , which decreases quadratically along trajectory lines near the origin, giving an extra ´2 weight ψ “ 1`y y2 damping in that region. We obtain „ B 2 βy0 psqψpy q ` aJ̌py qψpy q, pβy psqψpy qq “ ´3 ` 2i F̌ py q ` Bs 0 ψpy q 2 where the ` ψpy q in the brackets comes from the derivative hitting the numerator of ψpy q. The point here 2 is that the term in brackets is uniformly negative. For small y we have |2i F̌ py q| ď 2 and ψpy q is very small, while for high y the opposite happens. Doing the calculation for all intermediate values of y one concludes that the term in brackets is uniformly negative (for example, smaller than ´1{10). Note also that |aJ̌py qψpy q| ď Ca for some absolute constant C because |J̌| À y 2 near the origin (it has a second order zero at the origin). Therefore, letting Ωpsq “ βy0 psqψpηy0 psqq “ q̌py , sqψpy q, we conclude

B Ω “ f psqΩpsq ` E, Bs with some error |E| ď Ca and with f psq ď

´1 10 .

We also have that Ωp0q ď

(13) ε1 2

from our assumption (10).

Finally, choose a ! ε1 small enough and assume that |Ωpsq| ď ε1 is broken at time s0 , so we have Ωps0 q “ ε1 and Ω1 ps0 q ě 0 6 . Those assumptions directly give a contradiction from (13) at s “ s0 , provided 1 that 0 ą ´ 10 ε1 ` Ca, which can be ensured, because a ! ε1 . Finally, as |Ωpsq| ď ε1 , we obtain that }q̌p¨, sqψp¨q}L8 ď ε1 and therefore }q̌p¨, sq}L8 ď ε1 .

2.5 Conclusion Using Proposition 2.1, one concludes that }q}X remains uniformly bounded, provided q0 satisfies (10). For now on, let us fix those assumptions for q0 . Therefore, we have that ω “ F ` q solves the original OSW equation in the rescaled variables. Moreover, we know that s “ `8 corresponds to some finite dt time t “ T ˚ previous to the rescaling because ds “ λpsq and λ decays exponentially as one can see from (8), (12) and the fact that a is small enough. 5

This should not be surprising, as no term in M0 q has an a in front, so all the terms come from CLM, which admits a solution via complexification. 6 The same reasoning applies for Ωps0 q “ ´ε1 and Ω1 ps0 q ď 0.

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Theorem 2.3. There exist some values a0 , εi , δ, which can be chosen all small enough such that the following holds. Let a P p0, a0 s and ω0 pxq “ F pxq ` q0 pxq all odd, where q0 satisfies the bounds (10). Then, we have that the solution to OSW with initial data ω0 is given by ˆ ˆ ˙ ˆ ˙˙ 1 µptqx µptqx ωpx, tq “ F `q , sptq , λptq λptq λptq where q satisfies the global bounds (11) and λptq Ñ 0 as t Ñ T ˚ 7 . In particular, we have that both Hωp0, tq and ωx p0, tq blow up as t Ñ T ˚ . Remark. The assumptions on q0 can be formulated in terms of ω0 . Basically we require that ω0 is odd and close enough to the self-similar profile F . The assumption Hqp0q “ qy p0q “ 0 can be satisfied by modifying the modulation variables initial conditions λp0q, µp0q, so it does not impose a restriction on ω0 . Remark. As a final comment, let us note that our Theorem includes self-similar singularities for compactly supported smooth initial data. As the self-similar profile F decreases, one can choose q0 so that q0 pxq “ ´F pxq for x R r´R, Rs and R big enough, while still satisfying assumptions (10). Choosing q0 smooth also ensures that F ` q0 is smooth, so we can construct compactly supported and smooth initial data ω0 so that OSW blows up in a self-similar way.

3. Self-similar profiles for isentropic compressible Euler The other result of this article is joint work with Tristan Buckmaster and Javier Gómez-Serrano, and focuses on proving the existence of radial smooth self-similar profiles for the isentropic compressible Euler or Navier–Stokes. The equation is given by $ ρBt u ` ρu∇u “ ´∇p ` ν∆u, ’ ’ ’ & ρt ` divpuρq “ 0, (14) ’ ’ 1 ’ %ppρq “ ργ , γ where the viscosity ν “ 0 corresponds to Euler and ν ą 0 to Navier–Stokes. The first equation corresponds to the conservation of momentum (or equivalently, Newton’s second law). The second equation is the conservation of mass along trajectory lines and the third one is the isentropic law for the pressure of an ideal gas. The parameter γ is called adiabatic constant and we will center here in the monatomic ideal gas case which corresponds to γ “ 5{3. We will also restrict to the Euler case, that is ν “ 0. In the forthcoming paper [1], we extend the range of γ, we obtain other types of profiles and we use the Euler profiles to perform a stability analysis for the Navier–Stokes case, obtaining self-similar singularities for the isentropic compressible Navier–Stokes equations. This approach is inspired in the recent series of papers [20, 22, 21], which solved the outstanding problem of the existence of singularities for compressible Navier–Stokes with smooth initial data, for some values of γ. However, no smooth self-similar profiles were found for the monatomic gas (γ “ 53 ) before 7

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this work, so our result is completely new. The type of profiles found is also new, and in particular they concentrate faster than those found in [20].

3.1 Self similar equation First of all, we will always work with radial solutions to (14). We also impose ν “ 0, as we will work with the Euler equations. We write the equation in radial coordinates upRq and ρpRq, with u being a vector in 1 5 the radial direction and R “ |x|. Then, we define α “ γ´1 2 (which is 3 for γ “ 3 ) and perform the change of variables w pR, tq “ upR, tq ` α1 ρα and zpR, tq “ upR, tq ´ α1 ρα . This change of variables was proposed by Riemann [24] and thus pw , zq are usually called the Riemann invariants. The equation obtained for the Riemann invariants is ˆ ˙ $ 1`α 1´α α ’ ’ w` z BR w ` pw 2 ´ z 2 q “ 0, ’ &Bt w ` 2 2 2R (15) ˙ ˆ ’ ’ α 1`α 1´α ’ 2 2 %Bt z ` w` z BR w ´ pw ´ z q “ 0, 2 2 2R which exhibit a lot of symmetry. We are looking for self-similar solutions, and ` inspired ˘ by the two parameter 1 R R family of scaling symmetries of (15), we try the ansatz w pR, tq “ r T ´t W pT ´tq1{r (and the same for z). Here, r is just a parameter, so the factor 1r is just a constant factor that will make computations simpler. We also observe that the lower the r , the faster that the profile will expand, as the exponent 1r will be R bigger. Defining ξ “ pT ´tq 1{r , one obtains that the self-similar profiles W pξq, Z pξq satisfy $ NW pW , Z q ´rW ´ p1 ` 2αqW 2 {2 ` p1 ´ αqWZ {2 ´ αZ 2 {2 ’ ’ “ , ξB W “ ξ ’ & 1 ` p1 ` αqW {2 ` p1 ´ αqZ {2 DW pW , Z q ’ ’ NZ pW , Z q ´rZ ´ p1 ` 2αqZ 2 {2 ` p1 ´ αqWZ {2 ´ αW 2 {2 ’ %ξBξ Z “ “ , 1 ` p1 ´ αqW {2 ` p1 ` αqZ {2 DZ pW , Z q

(16)

which is an algebraic autonomous dynamical systems (the change ξ˜ “ logpξq makes the system autonomous) in two dimensions. Note that this dynamical system depends on two parameters γ and r , and that NW , NZ , DW , DZ are just polynomials in W and Z . This formulation has been known since Guderley [16], and it is not difficult to prove the existence of self-similar profiles of limited regularity C k from those equations. However, the question of existence of smooth solutions to (16) is much more difficult. We have plotted the phase portrait in Figure 3.1. The fundamental problem for the existence of smooth profiles is the singular point P2 , defined as the intersection of DZ “ 0 and NZ “ 0. The solutions we are looking for start at point P1 (point at infinity in the asymptotic direction p1, ´1q) at ξ “ 0, then pass smoothly through P2 at ξ “ 1 and end up reaching P3 “ p0, 0q at ξ “ `8. The profiles starting at P1 and ending at P3 ensure that the solution w pR, tq “ TR´t W pξq is non-zero at R “ 0 and does not grow as R Ñ `8. Even though the profile is singular at ξ “ 0, note that the real solution w pR, tq before rescaling is not, due to the factor R multiplying8 . 8

If one prefers profiles not singular at 0, one can equivalently work with ξW pξq as a profile and modify the rescaling appropriately.

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Figure 3.1: Phase portrait of (16) at r “ 1.1 and γ “ 5{3 at different scales. As one can see in the phase portrait, solutions going from P1 to P3 have to pass through the line DZ “ 0, which is a major problem in terms of the equation (16), as it corresponds to ξBξ Z blowing up. Thus, the only possibility for a solution crossing DZ “ 0 smoothly, is to cross through the point P2 , where both DZ and NZ cancel and the quotient may remain bounded.

3.2 Local analysis around P2 One can look at the modified dynamical system Wζ “ NW DZ and Zζ “ NZ DW , which just corresponds to an appropriate reparametrization in time of our original system (because pWξ , Zξ q is proportional to pWζ , Zζ q, so trajectories are locally preserved). The point P2 is an equilibrium point of the new system and we can perform a local analysis of this point looking at the eigenvalues of the Jacobian of pNW DZ , NZ DW q. ? Doing that, we see there exists a value r ˚ pγq ą 1 (which is 3 ´ 3 for γ “ 35 ) such that both eigenvalues of the Jacobian are positive for r P r1, r ˚ pγqq. Moreover, letting kpr q ě 1 to be the quotient of the eigenvalues, we have that kpr q is monotonically increasing from kp1q “ 1 to limr Ñr ˚ kpr q “ `8. The dynamical system theory tells us that P2 is a focus, and when k R N, generic solutions near P2 will have limited C k regularity as we tend to P2 . However, there will be two exceptional invariant curves of the system passing smoothly through P2 , which correspond to two smooth solutions of (16) through P2 . One of them agrees up to order tku with all the non-smooth trajectories, and we will focus on that one (which we simply refer as “the” smooth solution). Taking derivatives of our ODE (16) and evaluating at P2 , one can find a recurrence for the Taylor coefficients pWn , Zn q of the smooth solution. We can see that they define a convergent series continuous 1 ˇ with respect to the parameter r . More importantly, we get an expression for Zn of the form Zn “ n´kpr q Zn , where Zˇn is some polynomial expression in the previous coefficients. This is of paramount importance, because it indicates that the n-th Taylor coefficient blows up as kpr q Ñ n.

3.3 Statement and idea of the proof Theorem 3.1 (Buckmaster, Cao-Labora and Gómez-Serrano [1]). Let γ “ 5{3. There exists a value of r for which kpr q P p3, 4q, such that there is a smooth solution to (16) emanating from the point P1 at ξ “ 0, passing smoothly through P2 at ξ “ 1 and reaching P3 asymptotically as ξ Ñ `8.

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First of all, the existence of a solution emanating from P1 follows easily from calculating its Taylor series recurrence and showing convergence. The rest of the proof is divided in the two following propositions. Proposition 3.2. Let γ “ 5{3. There exists a value of r with kpr q P p3, 4q such that the smooth solution around P2 coincides with the unique solution emanating from P1 . This proposition is proved via a shooting argument in r . We prove that for kpr q “ 3`ε and kpr q “ 4´ε (for ε small enough) the solution around P2 stays respectively below and above the solution emanating from P1 . Continuity with respect to r ensures there is an intermediate value of r for which both solutions agree. In order to show the expected behaviour of the solution for k “ 3 ` ε and k “ 4 ´ ε we take advantage of the fact that |Z3 | and |Z4 | blow up, respectively, as ε Ñ 0. The argument is formalised via a concatenation of barrier arguments involving detailed local behaviour of the solution around P2 . Proposition 3.3. Let γ “ 5{3 and k P p3, 4q. We have that the smooth solution at P2 reaches point P3 at ξ “ `8. This proposition is also shown via barrier arguments. We also need to take into account the blow-up of the Taylor series as k is close to 3 or 4, however we need to prove the result for all the intermediate values as well. The strategy is to introduce a reparametrization in the barrier (singular as k approaches natural numbers) that desingularizes the barrier conditions. After desingularization, the barriers are proved via computer-assisted proofs. A recent example in using computer assisted-proofs for proving barrier arguments is [17], where self-similar profiles for a model of polytropic gaseous stars are obtained. A general survey of computer-assisted proofs in PDEs is [15].

Expression of gratitude I would like to thank Javier Gómez-Serrano for his outstanding work as my supervisor for my Degree Thesis and his collaboration in all the presented work. I would also like to thank Tristan Buckmaster for introducing to me the two problems discussed and for his fruitful collaboration in both of them. I am also very grateful to Federico Pasqualotto for our collaboration in the OSW model. I would like to thank the Princeton Department of Mathematics for their kind welcome and their support regarding the university fee in the development of this work. I am also grateful to the CFIS Mobility program funded by Fundació Privada Cellex for his partial support. I also received some financial support for my stay from the MOBINT program. I would like to thank Reports@SCM for the opportunity to publish this work as consequence of being awarded the Noether prize from the SCM. I am deeply honoured to receive this award and would like to thank the jury members and the SCM.

References [1] T. Buckmaster, G. Cao-Labora, J. Gómez-Serrano, “Smooth imploding solutions for 3D compressible fluids”, Forthcoming work (2021).

[2] T. Buckmaster, S. Shkoller, V. Vicol, “Formation of point shocks for 3D compressible Eu-

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ler”, Preprint (2019), https://arxiv.org/ abs/1912.04429.

transport equations”, Preprint (2019), https: //arxiv.org/abs/1906.05811.

[3] T. Buckmaster, S. Shkoller, V. Vicol, “Forma- [13] T.M. Elgindi, I.-J. Jeong, “On the effects of tion of shocks for 2D isentropic compressible advection and vortex stretching”, Arch. Ration. Euler”, Preprint (2019), https://arxiv.org/ Mech. Anal. 235(3) (2020), 1763–1817. abs/1907.03784. [14] C.L. Fefferman, “Existence and smoothness of [4] Á. Castro, D. Córdoba, “Infinite energy sothe Navier–Stokes equation”, The millennium lutions of the surface quasi-geostrophic equaprize problems 57 (2006), 67. tion”, Adv. Math. 225(4) (2010), 1820–1829. [15] J. Gómez-Serrano, “Computer-assisted proofs [5] J. Chen, T.Y. Hou, D. Huang, “On the fiin PDE: a survey”, SeMA J. 76(3) (2019), 459– nite time blowup of the De Gregorio model 484. for the 3D Euler equation”, Preprint (2019), https://arxiv.org/abs/1905.06387. [16] K.G. Guderley, “Starke kugelige und zylinrische verdichtungsstosse in der nahe des kugelmit[6] C, Collot, T.-E. Ghoul, S. Ibrahim, N. Masterpunktes bzw. Der zylinderachse”, Luftfahrtmoudi, “On singularity formation for the forschung 19 (1942), 302. two dimensional unsteady Prandtl’s system”, Preprint (2018), https://arxiv.org/abs/ [17] Y. Guo, M. Hadzic, J. Jang, M. Schrecker, 1808.05967. “Gravitational collapse for polytropic gaseous stars: self-similar solutions”, Preprint (2021), [7] C. Collot, T.-E. Ghoul, N. Masmoudi, “Sinhttps://arxiv.org/abs/2107.12056. gularity formation for Burgers equation with transverse viscosity”, Preprint (2018), https: [18] T. Kato, C.Y. Lai, “Nonlinear evolution equa//arxiv.org/abs/1803.07826. tions and the Euler flow”, J. Funct. Anal. 56(1) (1984), 15–28. [8] P. Constantin, P.D. Lax, A. Majda, “A simple one-dimensional model for the three- [19] J. Luk, J. Speck, “Shock formation in solutions dimensional vorticity equation”, Commun. Pure to the 2D compressible Euler equations in the Appl. Math. 38(6) (1985), 715–724. presence of non-zero vorticity”, Invent. Math. 214(1) (2018), 1–169. [9] A. Córdoba, D. Córdoba, M.A. Fontelos, “Formation of singularities for a transport equation [20] F. Merle, P. Raphaël, I. Rodnianski, J. Szefwith nonlocal velocity”, Ann. of Math. 162 tel, “On smooth self similar solutions to the (2005), 1377–1389. compressible Euler equations”, Preprint (2019), https://arxiv.org/abs/1912.10998. [10] A.-L. Dalibard, N. Masmoudi, “Separation for the stationary Prandtl equation”, Publ. Math. [21] F. Merle, P. Raphaël, I. Rodnianski, J. SzefInst. Hautes Études Sci. 130(1) (2019), 187– tel, “On the implosion of a three dimensional 297. compressible fluid”, Preprint (2019), https: //arxiv.org/abs/1912.11009. [11] S. De Gregorio, “On a one-dimensional model for the three-dimensional vorticity equation”, J. [22] F. Merle, P. Raphaël, I. Rodnianski, J. Szeftel, Stat. Phys. 59(5-6) (1990), 1251–1263. “On blow up for the energy super critical defocusing nonlinear Schrödinger equations”, In[12] T.M. Elgindi, T.-E. Ghoul, N. Masmoudi, “Stavent. Math. (2021), 1–167. ble self-similar blowup for a family of nonlocal

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Verlag der Dieterichschen Buchhandlung, 1860. [23] H. Okamoto, T. Sakajo, M. Wunsch, “On a generalization of the Constantin–Lax–Majda equation”, Nonlinearity 21(10) (2008), 2447. [25] T. Tao, “Lecture notes 4 for 247A”, https: //www.math.ucla.edu/~tao/247a.1.06f/. [24] B. Riemann, Über die Fortpflanzung ebener [26] E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, 1948. Luftwellen von endlicher Schwingungsweite,

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AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMÀTIQUES

Homotopical realizations of infinity groupoids ∗ Jan

McGarry Furriol

University of Copenhagen mcgarryjan@gmail.com ∗Corresponding author

Resum (CAT) La hipòtesi d’homotopia de Grothendieck afirma que l’estudi dels tipus d’homotopia dels espais topològics és equivalent a l’estudi dels ∞-grupoides. En la pràctica, un cop triat un model per a les categories d’ordre superior, l’equivalència és realitzada per l’assignació del ∞-grupoide fonamental a un espai topològic. Proposem un model accessible per al ∞-grupoide fonamental, usant categories topològiques per a modelitzar els ∞-grupoides.

Abstract (ENG) Grothendieck’s homotopy hypothesis asserts that the study of homotopy types of topological spaces is equivalent to the study of ∞-groupoids. In practice, after one has chosen a model for higher categories, the equivalence is realized by the assignment of the fundamental ∞-groupoid to a topological space. We propose an accessible model for the fundamental ∞-groupoid, using topological categories to model ∞-groupoids.

Acknowledgement I would like to thank Carles Casacuberta

Keywords: homotopy hypothesis, fundamental infinity groupoid, Moore paths. MSC (2010): 55U40, 55P15, 18B40. Received: July 31, 2021. Accepted: December 16, 2021.

http://reportsascm.iec.cat

for his invaluable support during his supervision of my bachelor thesis, on which this article is based, and of the project for which I was supported by grant 2019/COLAB of the Spanish Ministerio de Educación, Cultura y Deporte.

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1. Introduction This article results from the study of an equivalence between topological spaces and ∞-groupoids in the framework of homotopy theory: the homotopy hypothesis. This equivalence was described in the bachelor thesis of the author [7], and as done there, we now use it to give a more manageable model for the fundamental ∞-groupoid of a topological space in that context. We will after study when this assignment also gives such an equivalence.

What is algebraic topology? The fundamental concept in this area of mathematics is that of homotopy, which is a notion of continuous “deformation”. Concretely, two maps are homotopic if we can “deform” one into the other in a continuous manner. From here stems the notion of (weak) homotopy equivalence between topological spaces; for example, the plane without the origin is homotopy equivalent to the circle. The goal of algebraic topology is to classify the homotopy types of topological spaces, where we say that two spaces are of the same homotopy type if they are homotopy equivalent. To do this, invariants under homotopy equivalence are studied, such as the fundamental group. The fundamental group is a very useful invariant to classify homotopy types, but is far from allowing a complete classification.

Models for the homotopy types A more natural way of packaging the information of the fundamental group is to consider the fundamental groupoid, since we do not need to make any choice of basepoint. The fundamental groupoid is a category whose objects are the points of the topological space and whose morphisms are homotopy classes of paths. It is not surprising that this object does not permit us to model the homotopy types completely, because we discard a lot of information when considering the paths up to homotopy. Therefore we will consider paths as morphisms and keep the information about the homotopies separately. We will consider homotopies as 2-morphisms, that is morphisms between morphisms. In the same way, we will consider homotopies between homotopies as 3-morphisms, etc. The object obtained when taking into account all the higher homotopies is the so-called fundamental ∞-groupoid. This is a genuine idea of ∞-groupoid, with which Grothendieck suggested that there should exist a good model for the notion of ∞-groupoid for which the study of the homotopy types of topological spaces is equivalent to that of ∞-groupoids. This is what is generally known as the homotopy hypothesis, and it is seen as a proof for the suitability of a model of ∞-groupoids. The ideas of Grothendieck are found in his manuscript “À la poursuite des champs” [1], where the mathematician started a search for such a model.

Results Once equipped with some ideas in this introduction we can better make sense of the results involved in this article. We will model ∞-groupoids by certain topological categories. The way in which we proved in [7] the mentioned equivalence is by giving an equivalence of homotopy categories, a setting in which homotopy equivalences, or more generally weak equivalences, become isomorphisms. The result in [7] was:

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Theorem A. The homotopy category of topological spaces is equivalent to the homotopy category of ∞-groupoids, in the sense of Definition 3.3. This equivalence is realized by a composition of functors |C(Sing(−))| and its inverse |N(Sing(−))|, introduced in Section 5. This is based on the theory of model categories and the classical homotopy-theoretic equivalence of topological spaces with simplicial sets in [8]; Theorem 2.2.5.1 in [5] – also due to Joyal; and [4]. We will use this to proof our main result, which says that the model that we propose for the fundamental ∞-groupoid has the homotopy type it should have, according to Theorem A: Theorem B. The ∞-groupoid Π(X ) constructed in Construction 4.1 is weakly equivalent to |C(Sing(X ))| for every topological space X . Finally, we describe a situation in which the functor Π from the category of topological spaces to the category of ∞-groupoids gives an equivalence of homotopy categories. We obtained the following, based on the classical equivalence between connected pointed topological spaces and group-like E1 -spaces – see Section 6 for the notation: Theorem C. The fundamental ∞-groupoid functor Π and the classifying space functor B induce an ≥0 equivalence of categories Ho(Top≥0 ∗ ) ' Ho(∞-Grpd∗ ).

2. Categories with weak equivalences We start with a first hint about the existence of homotopy theory in more general contexts than topological spaces. A category with weak equivalences is a category C equipped with a class of morphisms W of C which contains all isomorphisms of C, and satisfy the two-out-of-three property: for f and g any two composable morphisms in C, if two of {f , g , g ◦ f } are in W , then so is the third. We will define the homotopy category of C by a universal property which can more generally described as follows. Let C be an arbitrary category and let S be a subclass of the class of maps of C. By the localization of C with respect to S we mean a category S −1 C together with a functor γ : C −→ S −1 C having the following universal property. For every s ∈ S, γ(s) is an isomorphism; given any functor F : C −→ D with F (s) an isomorphism for all s ∈ S, there is a unique functor θ : S −1 C −→ D such that θ ◦ γ = F . Definition 2.1. Let (C, W ) be a category with weak equivalences. Then the homotopy category of C is the localization of C with respect to the class W of weak equivalences and is denoted by γ : C −→ Ho(C). If it exists, the homotopy category is the category obtained from C by formally inverting all the weak equivalences. Proving existence can entail set-theoretical difficulties. To guarantee the existence one usually relies on extra structure, such as a model category structure. This is our case. For example, the category of compactly generated topological spaces, denoted by Top, has a model structure [3, §2.4]. The weak equivalences are the weak homotopy equivalences: those maps which induce an isomorphism on each homotopy group and a bijection between path-components.

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3. ∞-groupoids as topological categories We now enter the context in which we will define ∞-groupoids: enriched categories – see [5, A.1.3, A.1.4] for details. The cases that interest us are categories enriched over topological spaces and over simplicial sets. These are “categories” in which there is a notion of morphisms of every order. Definition 3.1. A topological category is a category enriched over Top, the category of compactly generated topological spaces. In a topological category C, for each pair of objects x and y we have a topological space MapC (x, y ); we think of its points as 1-morphisms, and of a path in it as a 2-morphism. In general, for n > 1, an n-morphism in C is a homotopy between two (n − 1)-morphisms inside the space MapC (x, y ). We denote the category of topological categories and topologically enriched functors by Cattop . We have a functor π0 : Cattop −→ Cat to ordinary categories by taking path-components of the morphism spaces. Continuing with the ideas in the previous paragraph, π0 has the effect of taking homotopy classes of 1-morphisms in C, and discarding all higher homotopies. For a topological category C, we call π0 C its homotopy category. Definition 3.2. Let F : C −→ D be a functor between topological categories. We say that F is a weak equivalence if the following conditions hold: • For every pair of objects x, y ∈ C, the induced map MapC (x, y )

/ Map (Fx, Fy ) D

is a weak homotopy equivalence of topological spaces. • Every object of D is isomorphic in π0 D to Fx, for some x ∈ C. The analogy with topological spaces is clear: the given definition is a weak version of the notion of equivalence of categories, in the same way in which weak homotopy equivalences are with respect to homotopy equivalences. We now turn our attention to the most central definition in our work. Recall that a groupoid is a category in which every morphism in invertible. The following characterizes those topological categories in which every morphism is invertible up to higher order morphisms or, in other words, in which every morphism has a homotopy inverse. Definition 3.3. Let C be a topological category. We say that C is an ∞-groupoid if π0 C is a groupoid. The condition on π0 C guarantees that for every 1-morphism f of C there exists a 1-morphism g , in the opposite direction, such that g ◦ f and f ◦ g are in the same path-component as the identity in the corresponding endomorphism spaces. In other words, every 1-morphism has a homotopy inverse. Every morphism of order greater than 1 in a topological category is a homotopy in some topological space, and therefore it is already invertible up to homotopy. The category of ∞-groupoids and topologically enriched functors will be denoted by ∞-Grpd; it is a full subcategory of Cattop . We may consider its homotopy category, with the weak equivalences being the ones defined in Definition 3.2, because of [4].

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Recall that a topological monoid is a monoid with respect to which the binary operation is a continuous map. A useful observation is that the structure of a topological category with one object is just the structure of a topological monoid on the space of endomorphisms of the single object; just as for ordinary categories and monoids. A characteristic property of ∞-groupoids, which will feature in the proofs of Theorems B and C, is that every connected ∞-groupoid is weakly equivalent to a group-like topological monoid, i.e., a topological monoid M such that π0 M is a group. The precise statement is the following: Lemma 3.4. Let G be a connected ∞-groupoid. Let x be an arbitrary object of G and consider the monoid of endomorphisms EndG (x) within G considered as a topological category with one object. Then the inclusion functor EndG (x) −→ G is a weak equivalence of topological categories. Proof. The inclusion functor is fully faithful because it induces the identity on the single mapping space. It is essentially surjective because π0 G is a connected groupoid.

4. A model for the fundamental ∞-groupoid The equivalence of Theorem A is realized by applying to a topological space X a certain composition of functors – which will be introduced in Section 5 – yielding an ∞-groupoid |C(Sing X )|; Theorem A informally says that this ∞-groupoid models the homotopy type of X . But this object is not very transparent nor manageable without familiarity with simplicial sets. In this section we propose a model Π(X ) for the fundamental ∞-groupoid, which we believe to improve on this, being more intuitive and accessible. It will be the goal of the next section to show that Π(X ) is weakly equivalent to |C(Sing(X ))|, thus showing that it is a faithful model. The construction of Π(X ) will be based on the following ideas. Let X be a topological space and x ∈ X be a basepoint. Consider the space of loops ΩX based at x, that is the space of continuous paths from the unit interval into X which begin and end at x, with the compact-open topology. We can consider a binary operation on it given by concatenation of loops, which one can achieve, for example, reparametrizing the two loops to concatenate so that we obtain a new map from the unit interval. This way of composing loops does not make the operation associative, and thus does not equip ΩX with a topological monoid structure. One way to accomplish this is by extending the space of parametrization for the paths so that we don’t have to reparametrize when concatenating. This is achieved by the construction we describe next. Consider the topological space ΩM (X , x) := {(f , r ) ∈ X R≥0 × R≥0 | f (0) = f (r ) = x, f (s) = f (r ) for s ≥ r }, with the product topology, where R≥0 = [0, ∞) has the Euclidean topology and X R≥0 – the space of continuous maps from R≥0 to X – has the compact-open topology. This space is known as the space of Moore loops. Notice that the usual loop space ΩX includes into ΩM X as the subspace of loops (f , r ) with parameter r = 1. In fact, the space of Moore loops deformation retracts onto ΩX , so they are homotopy equivalent spaces. We will now see how an associative composition may be defined on ΩM X by looking directly at the model we propose for the fundamental ∞-groupoid.

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Construction 4.1. Let X be a topological space. We let Π(X ) be the following topological category: • The objects are the points of X . • If x and y are two objects, we let MapΠ(X ) (x, y ) be the space of Moore paths from x to y : {(f , r ) ∈ X R≥0 × R≥0 | f (0) = x, f (r ) = y , f (s) = f (r ) for s ≥ r }. • If g is a path that begins where f ends, we define (g , s) ◦ (f , r ) as (f ∗ g , r + s), where ( f (t) if 0 ≤ t ≤ r , (f ∗ g )(t) = g (t − r ) if t ≥ r . • The constant path (cx , 0) as identity. Proposition 4.2. The topological category Π(X ) is an ∞-groupoid for every space X . Proof. Every Moore path (f , r ) has a homotopy inverse (g , r ), where g goes along the same trajectory as f in reverse. The assignment of the fundamental ∞-groupoid to a topological space is clearly functorial, and preserves weak equivalences. Thus we have a functor Π : Top −→ ∞-Grpd which induces a functor between homotopy categories. Example 4.3. Let X be a topological space, and x ∈ X a point. The space of endomorphisms of x within the ∞-groupoid Π(X ) is the space of Moore loops ΩM (X , x) based at x. Lemma 3.4 says that the inclusion functor ΩM (X , x) −→ Π(X ) is a weak equivalence. We know aim to prove that this model is accurate.

5. Realizing the fundamental ∞-groupoid After a short technical discussion we will reach the main result of this section: Theorem B. This discussion involves a comparison with a combinatorial version of topological spaces: simplicial sets. The combinatorial geometry of simplicial sets arises from the simplex category : the category whose objects are finite totally ordered sets, and whose morphisms are order-preserving functions between them. The simplex category is denoted by ∆. In it, there are certain maps, called face and degeneracy maps, which encode all the combinatorics. Let C be a category. A simplicial object in C is a functor ∆op −→ C. A simplicial object in the category of sets is called a simplicial set. A simplicial object in the category of topological spaces is called a simplicial space. We denote the category of simplicial objects in C by sC. Simplicial sets have a notion of geometric realization, which is a functor to the category of topological spaces; denoted by | − |. For example, the geometric realization of a simplicial set X is obtained by gluing standard n-simplices, one for each element of Xn , according to the data of the face and degeneracy maps. That is, we glue the spaces Xn × ∆n along the mentioned maps.

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More precisely, and a bit more generally, let C be a category with all colimits – in the sense of category theory, a reference for which is [6] – and suppose that we have a functor c : ∆ −→ C. Then, associated to this given data, we define the geometric realization as the left Kan extension of c along the Yoneda embedding ∆ −→ sSet. This generalizes to simplicial spaces and bisimplicial sets – that is simplicial objects in simplicial sets – giving functors to the categories of topological spaces and of simplicial sets, respectively. For example, if X is a simplicial space, then Xn is a topological space, and we glue the product spaces Xn × ∆n along the face and degeneracy maps. If the Xn have the discrete topology, i.e., they are just sets, then we recover the geometric realization of simplicial sets. Geometric realizations come with a right adjoint which we may call a nerve functor. For the geometric realization of simplicial sets, this functor would be the singular complex functor Sing – together with the geometric realization functor, these are the functors establishing the homotopy-theoretic equivalence between topological spaces and simplicial sets in [8] mentioned in the introduction. As a different example, the nerve of a category C is a simplicial set whose set of n-simplices is the set of sequences of n composable maps. If C is a topological or a simplicial category, then this construction clearly generalizes and gives are simplicial space or a bisimplicial set, respectively. Consider the composition of the nerve and realization functors Cattop

N

/ sTop

|−|

/ Top.

We call this composition the classifying space functor, and we denote it by B. The reason behind this name is the observation that if a topological category has only one object, then it is the same as a topological monoid; if furthermore it is groupoid, then it is a topological group. For details about this construction see [10]. Lemma 5.1. The classifying space functor B carries weak equivalences of topological categories to weak equivalences of topological spaces. Proof. The nerve functor clearly carries a weak equivalence of topological categories to degree-wise weak equivalences of simplicial spaces, i.e., to weak equivalences between the respective spaces of n-simplices, for every natural number n. Then one notes that the degeneracy maps of the simplicial space NC are closed cofibrations because points are closed, by the weakly Hausdorff condition in the definition of compactly generated topological space. Thus it follows from [11, A1] that the geometric realization of NF is a weak equivalence of topological spaces. Now, a result relating geometric realizations and nerves in the different contexts within which our main results fit. It will allow us to use the mentioned proof of Theorem A, which relies heavily on the theory of simplicial sets, to prove Theorem B. To state the following result, we introduce the simplicial analogue of topological categories. A simplicial category is a category enriched over the category of simplicial sets. The category of simplicial categories is denoted by Cat∆ . Theorem A is partly based on a homotopy theoretic equivalence given by a geometric realization C and nerve N pair of adjoint functors. This nerve is known as the homotopy coherent nerve, and it is a functor N : Cat∆ −→ sSet.

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Proposition 5.2. The following diagram commutes up to weak equivalence in Top: B

x

|−|

o Top O

o sTop O

N

Cattop

|−|

|−|

sSete o

|−|

ssSet o

N

Sing

Cat∆

N

Proof. The left square diagram commutes up to homeomorphism – a reference for this is [9, p. 94]. For the right square diagram, we have a degree-wise weak equivalence |N(Sing G)| −→ NG, for every ∞-groupoid G, which | − | carries to a weak equivalence in Top, as in Lemma 5.1. The top diagram is just how we defined B. The bottom diagram commutes up to homotopy by [2, Cor. 2.6.3]. Proposition 5.2 says, in particular, that the diagram below at the left commutes up to weak equivalence. o Top O

B

|−|

sSet o

N

Cattop

Sing

Cat∆

Top Sing

sSet

Π

/ Cattop O |−|

C

/ Cat∆

What the next result says is that the same holds for the diagram at the right, and by Theorem A, this proves: Theorem B. The ∞-groupoid Π(X ) is weakly equivalent to |C(Sing(X ))| for every topological space X . Proof. By Theorem A, it suffices to show that Π(X ) is weakly equivalent to |C(Sing X )|. We will show in the proof of Theorem C that there is a weak equivalence BΩX ' X −→ BΠ(X ) – here we choose any basepoint for X , as we don’t claim naturality of the equivalence. Thus, by Proposition 5.2, we have a weak equivalence X −→ |N(Sing Π(X ))|. Applying to it the functor |C(Sing −)| we obtain that |C(Sing X )| and Π(X ) are weakly equivalent, by the equivalence of categories of Theorem A. This confirms to us that Π(X ) is indeed an accurate model for the fundamental ∞-groupoid.

6. An equivalence of categories In this section we make additional assumptions on our topological spaces and ∞-groupoids, and then prove that the functors Π and B induce an equivalence between the corresponding homotopy categories. These assumptions are what we need in order to make use of the classical equivalence between connected pointed topological spaces and group-like E1 -spaces. Let Top∗ denote the category of pointed topological spaces, i.e., with a distinguished basepoint. Similarly, let ∞-Grpd∗ denote the category of ∞-groupoids with a distinguished object. Any model structure carries over to the pointed setting [3, Prop. 1.1.8]. In particular we can consider homotopy categories. Now ≥0 let Ho(Top≥0 ∗ ) and Ho(∞-Grpd∗ ) denote the full subcategories of the corresponding homotopy categories consisting of the connected objects.

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The fundamental ∞-groupoid functor Π and the classifying space functor B preserve weak equivalences. Thus they induce a pair of functors between the homotopy categories above, which is in fact an equivalence: Theorem C. The fundamental ∞-groupoid functor Π and the classifying space functor B induce an ≥0 equivalence of categories Ho(Top≥0 ∗ ) ' Ho(∞-Grpd∗ ). Proof. Observe that the inclusion functor of the topological monoid ΩM (X , b) of Moore loops, based at the basepoint b of a path-connected pointed topological space (X , b), into Π(X , b) is a weak equivalence. Since B preserves weak equivalences by Lemma 5.1, we have a natural weak equivalence BΩM (X , b) −→ BΠ(X , b) induced by the inclusion. The homotopy equivalence ΩM (X , b) ' Ω(X , b) reduces the proof to the classical equivalence between connected pointed topological spaces and group-like E1 -spaces – a reference for this is [11, Prop. 1.4]. On the one hand it gives a natural weak equivalence (X , b) ' BΩM (X , b) −→ BΠ(X , b), by the above. On the other hand, any connected pointed ∞-groupoid (G, b) is weakly equivalent to the topological monoid of endomorphisms M at its basepoint – a weak equivalence is given by the inclusion. The property of G being an ∞-groupoid then says that π0 M is a group. Then the cited result says that M ' ΩBM. As above, we then have natural weak equivalences (G, b) o

M ' ΩM B(G, b)

/ ΠB(G, b)

induced by the inclusions. Finally, notice that these weak equivalences give natural isomorpisms in the correspondent homotopy categories.

References [1] A. Grothendieck, “À la poursuite des champs”, Unpublished letter to Quillen (1983).

Springer Science & Business Media, New York, 2013.

[2] V. Hinich, “Homotopy coherent nerve in Deformation theory”, Preprint (2007), https: //arxiv.org/abs/0704.2503.

[7] J. McGarry Furriol, “Homotopical realizations of infinity groupoids”, Bachelor thesis, University of Barcelona, 2020.

[3] M. Hovey, Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 2007.

[8] D. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin, Heidelberg, 1967.

[4] A. Ilias, “Model structure on the category of small topological categories”, J. Homotopy Relat. Struct. 10(1) (2015), 63–70.

[9] D. Quillen, “Higher algebraic K-theory: I”, In: Higher K-theories, Lecture Notes in Mathematics 341, Springer, Berlin, Heidelberg, 1973, pp. 85–147.

[5] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, Princeton University Press, [10] G. Segal, “Classifying spaces and spectral sequences”, Publ. Math. Inst. Hautes Études Sci. 2009. 34 (1968), 105–112. [6] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, [11] G. Segal, “Categories and cohomology theories”, Topology 13(3) (1974), 293–312.

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Table of Contents

Factor analysis: Existence of solution to factor models Adrià Prior Rovira

1

Hirzebruch signature theorem and exotic smooth structures on the 7-sphere Guifré Sánchez Serra

11

Random walks on supersingular isogeny graphs Enric Florit Zacarı́as

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Self-similar profiles in Analysis of Fluids. A 1D model and the compressible Euler equations Gonzalo Cao-Labora

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Homotopical realizations of infinity groupoids Jan McGarry Furriol

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