Timeline for Any PL-homology-manifold is homotopy equivalent to a manifold
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2015 at 20:09 | comment | added | Igor Belegradek | I was merely explaining that even when $S$ is simply-connected, there is no way to extend an arbitrary smoothing of $X_0$ to $X$. There is a similar phenomenon when $S$ is not simply-connected: For sufficiently large $n$ (probably $n>3$), there is a map $\phi$ from the set of homology $n$-spheres to the set of smooth structures on $S^n$ such that $\phi(\Sigma^n)$ is standard if and only if $\Sigma^n$ boudns a contractible manifolds. I suggest looking at Kervaire's paper on homology spheres. | |
| Dec 22, 2015 at 15:37 | comment | added | ε-δ | You assume that S is simply connected and you should not. | |
| Dec 18, 2015 at 16:48 | comment | added | Igor Belegradek | It is correct, if the boundary of $X$ is a homotopy sphere, which is how I apply it. Let $X_0$ be $X$ with small ball removed. The boundary of $X_0$ has two components: $S$ and $S_0$, where $S_0$ is the standard sphere. By excision the inclusion $i: S_0\to X_0$ is a homology isomorphism, and since everything is simply-connected, $i$ is a homotopy equivalence. By duality the inclusion $j: S\to X_0$ is also a homology isomorphism. If $S$ is simply-connected, then $j$ is a homotopy equivalence. | |
| Dec 18, 2015 at 16:23 | comment | added | ε-δ | By the way, you say "removing a small ball from the interior of the contractible manifold gives an h-cobordism". Are you sure (I do not think it is correct). | |
| Dec 17, 2015 at 15:29 | comment | added | Igor Belegradek | You might be interested in arxiv.org/abs/1406.1735. | |
| Dec 17, 2015 at 15:08 | vote | accept | ε-δ | ||
| Dec 17, 2015 at 15:07 | comment | added | ε-δ | Thank you very much. By the way, the question was not revised — it was understood wrongly. | |
| Dec 16, 2015 at 21:58 | history | answered | Igor Belegradek | CC BY-SA 3.0 |