34
Jes´ us Gonz´alez and Peter Landweber
Section 2 of the twisted and untwisted cohomology groups of BG. Note that the last step requires knowing that, when Em is non-orientable (as determined in Proposition 3.2 and Remark 3.3), the twisted coefficients ! agree with those Zα used in Theorem 2.3. But such a requirement is Z a direct consequence of Theorem 3.5. Since the torsion-free subgroups of H ∗ (Em ) are easily identifiable from a quick glance at the E2 -term of the CLSS for the G-action on Vm+1,2 , only the torsion subgroups in Theorems 1.1 and 1.2 in dimensions m − 1, m, and m + 1
(17)
are lacking description in this argument. A deeper analysis of the CLSS of the G-action on Vm+1,2 (worked out in Sections 5 and 6 for G = D8 , and discussed briefly in Section 8 for G = Z2 × Z2 ) will give us (among other things) a detailed description of the three missing cases in (17) except for the (m + 1)-dimensional group when G = D8 and m ≡ 3 mod 4. Note that this apparently singular case cannot be handled directly with the torsion linking form argument in the previous paragraph because the connectivity of Vm+1,2 only gives the injectivity, but not the surjectivity, of the first map in the composite ∗
p (18) H m−1 (BD8 ; Zα ) −→ H m−1 (B(Pm , 2); Zα ) ∼ = H m+1 (B(Pm , 2)).
To overcome the problem, in Section 6 we perform a direct calculation in the first two pages of the Bockstein spectral sequence (BSS) of B(P4a+3 , 2) to prove that (18) is indeed an isomorphism for m ≡ 3 mod 4—therefore completing the proof of Theorems 1.1 and 1.2. ∗=
2
H ∗ (E2,D8 )
#2$
H ∗ (E4,D8 )
#2$ #1$ {2} #1$ #2$
H ∗ (E6,D8 )
#2$ #1$ {2} #2$ #4$ #2$ {2} #1$ #2$
H ∗ (E8,D8 )
#2$ #1$ {2} #2$ #4$ #3$ {4} #3$ #4$ #2$ {2} #1$ #2$
3
4
5
6
7
8
9
10 11 12 13 14
Table 1: H ∗ (Em,D8 ) ∼ = H ∗ (B(Pm , 2)) for m = 2, 4, 6, and 8 The isomorphisms in (16) yield a (twisted, in the non-orientable case) symmetry for the torsion groups of H ∗ (Em ). This is illustrated (for