Timeline for "Skew Cohomology" of a Space
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2011 at 5:06 | comment | added | John Klein | Ah yes, I see that. | |
| Mar 4, 2011 at 5:04 | vote | accept | John Klein | ||
| Mar 4, 2011 at 4:58 | answer | added | Mariano Suárez-Álvarez | timeline score: 5 | |
| Mar 4, 2011 at 4:54 | comment | added | Tom Goodwillie | That is, you can make $\Sigma_{n+1}$ act on $S_n(X)$ by permuting the basis (the singular simplices) as you describe, and then twist this by tensoring with the sign representation on $\mathbb Z$. The coinvariants $S_n(X)_{\Sigma_{n+1}}$ form a chain complex, and its $\mathbb Z$-dual is your complex of invariant cochains. But the $2$-torsion (which arose from the fact that some basis elements fixed by some odd permutations) had no effect on the dual. | |
| Mar 4, 2011 at 4:41 | history | edited | Tom Goodwillie | CC BY-SA 2.5 |
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| Mar 4, 2011 at 3:43 | comment | added | Eric Wofsey | If you consider chains on simplices which are oriented but whose vertices are unordered and declare that the same simplex with opposite orientation is the negative of the original simplex, this should be exactly the same as taking $\Sigma_n$-coinvariants on the usual chain group. | |
| Mar 4, 2011 at 3:33 | history | edited | John Klein | CC BY-SA 2.5 |
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| Mar 4, 2011 at 3:32 | comment | added | John Klein | My recollection is that Lefschetz failed to order the vertices of his simplices. This is where the 2-torsion problems arise. But I don't see how this is dual to my construction... | |
| Mar 4, 2011 at 2:36 | comment | added | Eric Wofsey | I think I remember reading somewhere that the dual construction on homology was actually the first version of singular homology that was defined, but it was awkward to work with because the chain groups had 2-torsion. | |
| Mar 4, 2011 at 2:17 | answer | added | Eric Wofsey | timeline score: 10 | |
| Mar 4, 2011 at 0:34 | history | asked | John Klein | CC BY-SA 2.5 |