Timeline for What is the top cohomology group of a non-compact, non-orientable manifold?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2021 at 0:30 | comment | added | Tom Goodwillie | Andy, you're probably right that it was Stallings. I learned this fact from John Klein. Sketch proof: Suppose $H^{\>d}(X;M)=0$ for every $\mathbb Z[\pi_1(X)]$-module $M$. Make a $(d-1)$ dimensional complex $K$ with $(d-1)$-connected map $f:K\to X$. The relative homology $H_\ast(f;\mathbb Z[\pi_1(X)])$ vanishes below degree $d$. Because of the cohomology hypothesis, $H_d(f;\mathbb Z[\pi_1(X)])$ is the only nontrivial group and is a projective module. Attach enough $d$-cells to $K$ (trivially) to make that module free. A basis for it tells you how to attach $d$-cells to finish the job. | |
| May 27, 2021 at 18:21 | comment | added | Georges Elencwajg | Thank you for the amazing information contained in Whitehead's theorem, and for your note, Andy. | |
| May 27, 2021 at 17:19 | comment | added | Andy Putman | @TomGoodwillie: I believe that you're right on both counts, which gives an alternate approach to Whitehead's theorem except maybe in very low dimensions. If I remember correctly, the result you're referring to in your first comment is due to Stallings, right? I remember seeing the paper long ago, but never got around to reading it. In any case, one nice thing about Whitehead's proof is that it gives a very explicit and visually appealing deformation retract onto an (n-1)-dimensional spine. | |
| May 27, 2021 at 16:34 | comment | added | Tom Goodwillie | Also, I believe that $H^p(M;G)$ is isomorphic to $H_{n-p}(M;G\otimes \mathbb Z^t)^{lf}$(locally finite homology twisted by orientation), and that $H_0(M)^{lf}$ with any twisted or untwisted coefficients is pretty clearly zero for a connected non-compact manifold $M$. | |
| May 27, 2021 at 3:07 | comment | added | Tom Goodwillie | Conversely, it's also true that vanishing of $H^{\ge n}(X)$ for all local coefficient systems implies that $X$ is homotopy equivalent to a cell complex of dimension $<n$ (except maybe for very small $n$). | |
| May 27, 2021 at 2:35 | history | answered | Andy Putman | CC BY-SA 4.0 |