Timeline for Subvarieties with different topology representing the same cycle
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 18, 2013 at 19:05 | vote | accept | calc | ||
| Jun 18, 2013 at 19:05 | vote | accept | calc | ||
| Jun 18, 2013 at 19:05 | |||||
| Jun 17, 2013 at 20:51 | comment | added | pinaki | Ehresmann's Theorem and theorems of Varchenko (in the algebraic case) and Verdier in analytic (and possibly more general categories) case give a somewhat positive statement: if you take a "family" of homologous cycles, then "almost all" of them will be homeomorphic. | |
| Jun 17, 2013 at 20:25 | comment | added | calc | I am sorry, I forgot to include that the represented cycle is not the zero cycle. Does the question look better now? | |
| Jun 17, 2013 at 20:23 | history | edited | calc | CC BY-SA 3.0 |
added 10 characters in body; deleted 10 characters in body; added 30 characters in body
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| Jun 17, 2013 at 19:31 | answer | added | Danny Ruberman | timeline score: 1 | |
| Jun 17, 2013 at 19:29 | comment | added | Mark Grant | The answer to the question in your second paragraph is 'no'. For any $X$ of dimension at least $n$, one can always find non-homeomorphic submanifolds $Y$ and $Z$ of dimension $n-1$ which represent the zero element in $H_{n-1}(X)$. Perhaps bordism theory offers a context in which to formulate such questions (and give negative answers, I fear). | |
| Jun 17, 2013 at 19:24 | comment | added | G.C. | Consider all the dim $k$ submanifolds of the $n$-sphere for $k<n$ ... | |
| Jun 17, 2013 at 18:51 | history | asked | calc | CC BY-SA 3.0 |