Timeline for Noncompact homology spheres?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2016 at 2:50 | comment | added | Denis Nardin | @KestutisCesnavicius The proof in Hatcher uses only the local homology, so I believe that this answer covers also your more general case | |
| Apr 22, 2010 at 18:22 | comment | added | Paul | I looked up the math reviews of the Bryant-Ferry-Mio-Weinberger article and they say finite dimensional ANR, but not local contractibility. Perhaps it follows, I'm not sure. I'm also confused by what homological dimension means here: which flavor? perhaps all maps to K(Z,k) are nullhomotopic for k>d? | |
| Apr 22, 2010 at 14:14 | comment | added | Bill Kronholm | @Paul: I'm not an expert in homology manifolds, but I think the dimension that is typically used is homological dimension, or cohomological dimension. Maybe this condition and local contractibility (or local compactness) are only used in situations where they are needed and are not part of the general definition. | |
| Apr 22, 2010 at 5:49 | comment | added | Kestutis Cesnavicius | Same non-local homology, i.e., $H_i(X, X - x) = H_i(\mathbb{R}^n, \mathbb{R}^n - \{0\})$ and $H_i(X) = H_i(\mathbb{S}^n)$. And as Paul remarks, I have no further assumptions (except simply-connectedness if needed, though that probably has nothing to do with all of this). | |
| Apr 21, 2010 at 23:49 | comment | added | Paul | Are you sure that finite dimensional and locally contractible is part of the definition of homology manifold? What notion of dimension is being used here? I vaguely recall that the definition only required $H_i(X,X-x)=H_i(R^n,R^n-0)$ for all $x$, and that most non-manifold examples were not locally compact. | |
| Apr 21, 2010 at 20:51 | comment | added | Bill Kronholm | So, do you want a homology $n$-manifold with the same local homology as $S^n$ or the same (non-local) homology as $S^n$? | |
| Apr 21, 2010 at 20:08 | comment | added | Kestutis Cesnavicius | $R^n$ is non-compact but it does not have same homology as the $n$-sphere. | |
| Apr 21, 2010 at 15:07 | comment | added | Bill Kronholm | I edited to include a response to the second question. Does it help? | |
| Apr 21, 2010 at 15:01 | history | edited | Bill Kronholm | CC BY-SA 2.5 |
added response to second question.
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| Apr 21, 2010 at 13:25 | comment | added | Kestutis Cesnavicius | Thanks! How about generalized homology spheres? | |
| Apr 20, 2010 at 17:04 | history | edited | Bill Kronholm | CC BY-SA 2.5 |
fixed grammar
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| Apr 20, 2010 at 16:37 | history | answered | Bill Kronholm | CC BY-SA 2.5 |