Timeline for Inverse cohomological isomorphisms
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 5, 2014 at 4:52 | comment | added | ThiKu | Ah, I just see this is essentially already contained in Hatcher's answer below. | |
| Sep 5, 2014 at 4:49 | comment | added | ThiKu | By the way, for the example of the homology spheres, you don't need Quillen's plus construction to construct a continuous map realising the homology isomorphism. Just take a disk inside $M$, map its complement by a constant map to the north-pole $P$ of $S^3$ and the disk homeomorphically to $S^3-\left\{P\right\}$. This map has degree $1$, so induces an isomorphism of 3rd homology (and trivially of all other homology groups). | |
| Sep 4, 2014 at 18:52 | comment | added | Ben Wieland | The two hypotheses that the map be an isomorphism on fundamental group and an isomorphism of cohomology with trivial coefficients are not sufficient. There is an $S^n\times S^n$ bundle over $S^1$ with a map to $S^1\times S^{2n}$ that is an isomorphism on cohomology. | |
| Sep 4, 2014 at 15:01 | vote | accept | Włodzimierz Holsztyński | ||
| Sep 4, 2014 at 14:43 | answer | added | Allen Hatcher | timeline score: 7 | |
| Sep 4, 2014 at 9:51 | answer | added | Matthias Wendt | timeline score: 5 | |
| Aug 21, 2014 at 20:22 | comment | added | Qiaochu Yuan | Aha, great. I was thinking about Poincare dodecahedral space. | |
| Aug 21, 2014 at 17:44 | comment | added | Matthias Wendt | @QiaochuYuan: here is the argument I had in mind. Take an integral homology 3-sphere $M$ with contractible universal cover. Such things can be found among the Brieskorn manifolds. Then the map to the plus-construction $M\to M^+\simeq S^3$ is a continuous map which induces an isomorphism in cohomology. But any map $S^3\to M$ must have degree 0, so there can not be an inverse on cohomology. | |
| Aug 21, 2014 at 16:27 | comment | added | Qiaochu Yuan | @Matthias: I was trying to see if homology spheres gave a counterexample without the hypothesis of simple connectedness but I couldn't finish it. Can you give some details? | |
| Aug 21, 2014 at 6:16 | comment | added | Włodzimierz Holsztyński | Thank you Matthias and Eric. Furthermore, instead of simple-connectedness I could assume that $\ f\ $ induces also an isomorphism of the fundamental groups. | |
| Aug 21, 2014 at 6:11 | comment | added | Eric Wofsey | Smoothness is not necessary; every (paracompact Hausdorff) topological manifold has the homotopy type of a CW-complex. | |
| Aug 21, 2014 at 6:03 | comment | added | Matthias Wendt | Should be true (at least if the manifolds are smooth). Morse theory shows that the manifolds have CW-structures. Then Whitehead's theorem states that $f$ is a homotopy equivalence, hence there is an inverse homotopy equivalence. As you probably noticed, homology spheres show that simply-connected is necessary. | |
| Aug 21, 2014 at 5:54 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
a missed assumption included
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| Aug 21, 2014 at 5:15 | history | asked | Włodzimierz Holsztyński | CC BY-SA 3.0 |