The space-time interpretation of Poincare’s conjecture
(proved by G. Perelman)
Vasil Penchev Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models vasildinev@gmail.com
“Space And Time: An Interdisciplinary Approach� 3rd annual conference September 26, 2019 - September 28, 2019 Institute of Philosophy, Vilnius University: 201, Universiteto 9/1, Vilnius, Lithuania
Project: Đ”Đ? 15/14 - 18.12.2017 to the "Scientific Research" Fund, Bulgaria: "Non-Classical Science and Non-Classical Logics. Philosophical and Methodological Analyses and Assessments"
Contains: Background and prehistory A generalization of PoincarÊ’s conjecture Physical interpretation in terms of special relativity An idea for proving the generalization
Background and prehistory
“Poincaré’s conjecture” The French mathematician Henri Poincaré offered a statement known as “Poincaré’s conjecture” without a proof: He wrote: “Mais cette question nous entraînerait trop loin” - as the last sentence of his paper: Poincaré, H. (1904) "Cinquième complément à l'analysis situs" Rendiconti del Circolo Matematico di Palermo 18, 45-110.
“Poincaré’s conjecture” Poincaré formulated the problem in §6 (pp. 99-110) of his Fifth supplement to “Analysis Citus”: Meaning the context, he questioned: “Est-il possible que le groupe fondamental de V se réduise ál la substitution identique, et que pourtant V ne soit pas simplement connexe?”
The main source [MS] further Carlson, J. (ed.) 2014. The Poincaré conjecture: Clay Mathematics Institute Research Conference, resolution of the Poincaré conjecture, Institute Henri Poincaré, Paris, France, June 8-9, 2010. Clay math. proc.: vol. 19. Providence (RI): AMS for CMI It is freely downloadable from: http://www.claymath.org/library/proceedings/cmip19.pdf
A century of failures to be proved ... Anyway, nobody managed neither to prove nor to reject rigorously the conjecture about one century: Morgan, J.W. 2014. 100 Years of Topology: Work Stimulated by Poincaré’s Approach to Classifying Manifolds. In MS, pp. 7-29 (esp. pp. 19-25) “2.8. Why the Poincaré Conjecture has been so tantalizing.”
One of the seven “Millennium Problems” It was included even in the Millennium Problems by the Clay Mathematics Institute and a prize of $1,000,000 for its solution: https://www.claymath.org/millennium-problems “In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold.”
CMI’s popular representation “We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere.”
Grigoriy Perelman’s proof It was proved by Grigoriy (Grigori, Grigory, Grisha) Perelman in 2002-2003 published in Arxiv (freely downloadable): “The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman” (Press Release of March 10, 2010. In MS, pp. vii-xv: vii)
Perelman’s “Нет!” “He said nyet to $1 million. Grigory Perelman, a reclusive Russian mathematics genius who made headlines earlier this year for not immediately embracing a lucrative math prize, has decided to decline the cash” (Ritter, M. 1.7.10. "Russian mathematician rejects $1 million prize" https://phys.org/news/2010-07-russian-mathematici an-million-prize.html )
Perelman’s explanation “But the Interfax news agency quoted Perelman as saying he believed the prize was unfair. Perelman told Interfax he considered his contribution to solving the Poincare conjecture no greater than that of Columbia University mathematician Richard Hamilton” (Ritter, M. July 1, 2010, see the previous slide)
Perelman’s explanation in original "Если говорить совсем коротко, то главная причина - это несогласие с организованным математическим сообществом. Мне не нравятся их решения, я считаю их несправедливыми. Я считаю, что вклад в решение этой задачи американского математика Гамильтона ничуть не меньше, чем мой" (https://www.interfax.ru/russia/143603 )
A generalization of Poincaré’s conjecture
From a “3-sphere” to a “4-ball” One can mean not only a “3-sphere” in 4-Euclidean space, but the “internality” furthermore: a “4-ball” bounded by the 3-sphere The 4-ball is more relevant for the intended physical interpretation by Einstein’s special relativity and Minkowski space as its mathematical model
More about the generalization If one “unfolds” a 4-ball in 3 dimensions, the 3-sphere would be “unfolded” in 2 dimensions The 2-dimensional boundary of the 3-dimensional unfolding of the 4-ball would be the 3-sphere unfolded Further, we mean the generalization as to the space-time physical interpretation
Our space as a “4-ball”, topologically The generalization would state that any 4-dimensional ball (or “3-sphere”) is equivalent to 3-dimensional Euclidean space topologically: Though Euclidean space is open “by itself”, as if it can be “closed” in an additional, fourth dimension at least in a topological sense is what the generalized conjecture supposes
What the generalization means A continuous mapping exists so that it maps the former ball into the latter space one-to-one A visualization: one might deform gradually (i.e. mathematically continuously, or even smoothly) the closed ball transforming it into the open space
Seemingly paradoxical At first glance, that seems to be too paradoxical for a few mismatches The suggested visualization might illustrate mismatches forcing the conjecture to seem paradoxical: One might cancel the fourth dimension of the ball gradually transforming it into the openness of the space
Both mismatches The former is 4-dimensional and as if “closed” unlike the latter, 3-dimensional and as if “open” according to common sense The conjecture “equates” the two misfits in a way to compensate each other absolutely at least topologically Openness is equivalent to a new dimension closed
Discreteness and continuity So, any mapping seemed to be necessarily discrete to be able to overcome those mismatches ... ‌ and being discrete, this implies for the conjecture to be false As if: continuity in an additional dimension and discreteness without it might be the same ...
Physical interpretation in terms of special relativity
A unit 4-ball unfolded in 3 dimensions One may notice that the 4-ball is almost equivalent topologically to the “imaginary domain” of Minkowski space in the following sense of “almost”: That “half” of Minkowski space is equivalent topologically to the unfolding of a 4-ball
Unfolding a 4-ball One would obtain a 3-dimensional unfolding of a 4-ball as follows: 1. Cutting the ball by a 3-dimensional “knife” in an arbitrary 3-dimensional (usual) ball 2. A (well-)ordered set of 3-balls parameterized from “-r” to “+r” (where “r” is the radius of both 4-ball and 3-ball is the unfolding at issue)
The case of infinite radius ... Then, the unfolding would be an ordered set of 3-balls Further, each ball is topologically equivalent to the internality (i.e. without the surface) of a “finite” ball with its radius equal to its parameter The unfolding of an “infinite” 4-ball is topologically equivalent to any domain of Minkowski space without the light cone
What remains 1. The “knife” as to the topological difference of an “infinite” 4-ball and its 3-unfolding 2. The “light cone” as to the “half” of Minkowski space Might they be the same? Topologically, obviously yes!
To whom it was not obvious The light cone is an ordered set of spherical surfaces parametrized by its radius from “minus infinity” to “plus infinity” So, the light cone in turn is a 2-unfolding of the 3-”knife” assigned to a corresponding ball by the same parameter (= their radius) after unfolding
An idea for proving Poincare’s conjecture Indeed, here is a series of topological equivalences: 1. The “knife” space “is” Euclidean space 2. The light cone “is” the “knife” space 3. The 3-sphere “is” the light cone 4. Consequently, the 3-sphere “is” Euclidean space “Is” means ‘is topologically equivalent to’
The idea seen by elastic deformations 1. One “pulls” Euclidean space in the fourth dimension transforming it into the “light cone” of the future: the point of pulling is the present 2. One “deforms” that light cone to an infinite 3-sphere (or 3-hemisphere): the deformation is continuous 3. One “shrinks” the infinite 3-sphere to a unit one
The physical meaning of the generalization Then, the generalization means the topological equivalence of the physical 3-space and its model in special relativity In turn, that topological equivalence means their equivalence as to causality physically Anyway, causality is irreversible, and continuity is not
The irreversibility of causality Time’s arrow is what implies the irreversibility of cause and effect However, Minkowski space as the model of special relativity means time to be space-like and thus reversible just as continuity is Time’s arrow is a consistent complement to Minkowski space within special relativity
“Time’s arrow” Time’s arrow is absent in both Minkowski space and generalized conjecture It is represent in both by continuity It can be added consistently to both as a complemental and restricting condition if one considers each of them as a mathematical construction relevant to special relativity
Causality as topological equivalence Causality means continuity in a physical sense, and the topological equivalence conserves it Indeed, Đ° continuous series of shrinking neighborhoods links the cause to the effect That is the set-theory and topological interpretation of a continuous series of logical implications between them
In other words ... So, Grisha Perelman proving Poincare’s conjecture has proved furthermore the adequacy of Minkowski space as a model of the physical 3-dimensional space rigorously A model containing any topological mismatch would not conserve causality: that causal violation would reject special relativity
A mathematical proof of causality Of course, all experiments confirm the same empirically, but not mathematically as Perelman did Perelman’s proof excludes any experimental refusal in future and in principle as to causality Furthermore and rather shocking: the other “half” of Minkowski space is not less relevant causally
An idea for proving the generalization
“Unfolding” the problem Topologically seen, the problem turns out to be reformulated so: One needs a proof of the topological equivalence of the “infinite” 4-ball and its unfolding by 3-balls + the “knife”: That is what the “half” of Minkowski space is, topologically
“Unfolding” the problem (2) Poincare’s conjecture means a finite or “unit” 3-sphere rather than an “infinite” one Its generalization means an infinite 4-ball in general and implicitly, an infinite 3-sphere The finite and infinite ones of the same kind are equivalent topologically to each other Thus, the mismatch is not essential
Indeed ... The meant “half” of Minkowski space is equivalent to a continuous interval of Euclidean spaces The number of its elements is: “infinity (for the “unfolding”) plus one (for the “knife”)” A continuous “interval” of Euclidean spaces is equivalent topologically to a single one as both are continuous and their set theory power is the same
Indeed (2) ... One can call “not-knife” that single Euclidean space (topologically equivalent to the “unfolding”) What remains to be proved is the topological equivalence of both “not-knife” and “knife” Euclidean spaces (discrete to each other) to an Euclidean space
Indeed (3) ... One can divide Euclidean space into two disjunctive subspaces: For the example: by the parameter of any dimension: the one, “less and equal than any constant of the parameter”; the other, “greater than it” Homeomorphism refers only to open subsets and can ignore all closed sets containing the border
Indeed (4) ... Then, the union of both disjunctinctive subspaces is the Euclidean space itself Each of the subspaces is topologically equivalent to one of the “knife” and “not-knife” Euclidean spaces correspondingly
Conclusion Consequently, the “knife” and “not-knife” Euclidean spaces are equivalent topologically to one single Euclidean space An idea about proving the generalization of Poincaré’s conjecture is sketched An idea for the Poincaré conjecture itself was sketched a few slides ago in the same framework
Thank you for your kind attention!
Any questions or comments are welcome!