Timeline for Relations between Betti numbers for clique complex
Current License: CC BY-SA 3.0
6 events
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| Feb 26, 2018 at 19:54 | comment | added | Henry.L | @PeterHeinig Thanks! Sure, but for a more general simplicial complex I am not even sure in that case if two consecutive Betti numbers are related since there could be much more possibilities than a clique complex. | |
| Feb 26, 2018 at 19:13 | comment | added | Peter Heinig | Since you ask "are there further...relations between $\beta_k$ and $\beta_{k+1}$, and since the inequality you are citing holds for any simplicial complex (not only for clique complexes of the 1-skeleton), it is not off-topic to mention the usual formula, also valid in general and not specific to your setting, for the Euler-characteristic: $f_0-f_1+\dotsm+(-1)^{\mathrm{dim}(K)} f_{\mathrm{dim}(K)} = \beta_0 - \beta_1 + \dotsm + (-1)^{\mathrm{dim}(K)}\beta_{\mathrm{dim}(K)}$. Again, this is a validity on the class of all abstract simplicial complexes, not specific to your question. | |
| Feb 26, 2018 at 18:34 | comment | added | Benjamin Steinberg | The barycentric subdivision of a simplicity complex is always a flag complex and this doesn't change the homology. | |
| Feb 26, 2018 at 18:27 | history | edited | Henry.L | CC BY-SA 3.0 |
edited body
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| Feb 26, 2018 at 18:20 | history | edited | Henry.L | CC BY-SA 3.0 |
added 10 characters in body; edited title
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| Feb 26, 2018 at 18:15 | history | asked | Henry.L | CC BY-SA 3.0 |