Timeline for Realization problem for Betti numbers
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2018 at 10:01 | answer | added | Michael Albanese | timeline score: 9 | |
| Jul 6, 2018 at 10:01 | history | edited | Michael Albanese | CC BY-SA 4.0 |
added 21 characters in body; edited tags
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| Jun 3, 2013 at 12:50 | comment | added | Lennart Meier | Regarding the question when the only non-zero Betti numbers are $0$, $2k$ and $4k$: If we want the stronger statement that $H_i(M) = \mathbb{Z}$ if $i = 0,2k,4k$ and $0$ else, then the only possibilites are indeed the dimensions of the complex, quaternionic and octonionic plane, i.e. $4k = 4,8$ or $16$. For rational coefficients there are more possibilities, e.g. $4k = 32$, but a general restriction is that $k$ has to be even (and not $24$) if $M$ is simply-connected. You might have a look at this article by Su: arxiv.org/pdf/1010.3274v1.pdf | |
| May 31, 2013 at 20:45 | comment | added | Liviu Nicolaescu | The minimal $b_6$ you seek is $2$. The manifold $S^6\times S^6$ has Poincare polynomial $1+2t^6+t^{12}$. | |
| May 31, 2013 at 20:43 | vote | accept | Julien Marché | ||
| May 31, 2013 at 20:41 | comment | added | Liviu Nicolaescu | There are similar obstructions in dimension 20. Check my answer to this question mathoverflow.net/questions/116814/… | |
| May 31, 2013 at 20:22 | answer | added | nsrt | timeline score: 4 | |
| May 31, 2013 at 17:01 | history | asked | Julien Marché | CC BY-SA 3.0 |