Timeline for Ordinal-indexed homology theory?
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2011 at 4:38 | comment | added | Pete L. Clark | @Jim: according to my definition, it comes out to be the same as $H_{\omega}(X)$: i.e., the same answer for all infinite ordinals. (I said it was a stupid definition...) | |
| Feb 13, 2011 at 13:58 | comment | added | Jim Conant | @Pete: Suppose we take that as a definition. How would we define $H_{\omega+1}(X)$? | |
| Feb 13, 2011 at 7:20 | comment | added | Pete L. Clark | (In the above comment, "undetermined" |-> "underdetermined". I am too lazy to do the retexing necessary to fix this.) | |
| Feb 13, 2011 at 7:19 | comment | added | Pete L. Clark | Wow, this time we got an accepted answer too quickly for my taste. I don't know about Floer homology (except that my grad school roommate was a big fan), but the more basic question asked still seems interesting...but undetermined. Can anyone say anything about what properties we want an ordinal-indexed homology theory to satisfy? I mean, here is something stupid: for every infinite ordinal $\alpha$, put $H_{\alpha}(X) = \bigoplus_{i < \aleph_0} H_i(X)$. Doubtless this is not what we want, but why ? | |
| Feb 13, 2011 at 2:17 | comment | added | Tim Perutz | In Floer theories, two generators for the cohomology have a relative degree, the index of a certain Fredholm operator. However, ordinals don't really give a good picture because the Fredholm operator is not uniquely determined, and its index is only uniquely determined mod some $N$; the relative degree then lies in $\mathbb{Z}/N$. | |
| Feb 13, 2011 at 2:12 | comment | added | Tim Perutz | Floer cohomology theories (there are several by now, with many applications) are in principle "semi-infinite cohomology spaces" for polarised Hilbert manifolds. The work of Lipyanksiy noted in Kevin Walker's answer gives an approach which is unusual in that it doesn't invoke differential equations. It's still analytic, though (he makes essential use of the interplay between weak and strong topologies on Hilbert space). | |
| Feb 13, 2011 at 0:51 | comment | added | Qiaochu Yuan | @Tilman: okay, so for every alpha there is an "alpha-simplex" category (the category of nonempty ordinals less than alpha), and "alpha-homological algebra" ought to be the study of "alpha-simplicial" abelian groups. So the obvious question here is whether some form of Dold-Kan holds. | |
| Feb 13, 2011 at 0:00 | vote | accept | Jim Conant | ||
| Feb 12, 2011 at 23:07 | answer | added | Kevin Walker | timeline score: 4 | |
| Feb 12, 2011 at 22:36 | comment | added | Jim Conant | @Martin: I agree that the usual simplex approach doesn't generalize. | |
| Feb 12, 2011 at 22:00 | comment | added | Tilman | Have you looked at Po Hu's paper on transfinite spectral sequences? You might get some inspiration there. No transfinitely indexed homology theories there, though. You could also ask if there is transfinite homological algebra. The ordinary context with chain complexes doesn't seem to generalize in some obvious way, but the simplicial setting might? | |
| Feb 12, 2011 at 21:58 | comment | added | Marty | I seriously doubt it. Just from the algebraic standpoint, if $\alpha$ is a limit ordinal, then I'd bet that $H_\alpha$, $H_{\alpha + 1}$, $H_{\alpha + 2}$, etc.. would have to form an ordinary homology theory, and it would be impossible to connect $H_\alpha$ to $H_\beta$ for any ordinal $\beta < \alpha$. | |
| Feb 12, 2011 at 21:44 | comment | added | Martin Brandenburg | Also, in order to make other properties hold, the usual approach with simplices cannot be generalized, unless you introduce infinite sums of simplicies whose support is restricted by some ordinal. But I doubt that this works at all. | |
| Feb 12, 2011 at 21:41 | comment | added | Martin Brandenburg | Interesting idea, but is it possible to extend the long exact pair sequence $H_{\omega}(A) \to H_{\omega}(X) \to H_{\omega}(X,A) \to H_{\omega+1}(A) \to ...$ to the left in a reasonable way? | |
| Feb 12, 2011 at 21:41 | comment | added | Peter LeFanu Lumsdaine | I’ve no idea, but: are there any obvious examples of spaces (or algebras, etc.) for which one would expect transfinite-dimensional cohomology to give interesting information? Without that, this is a cute question; with that, it becomes a very interesting one indeed… | |
| Feb 12, 2011 at 21:08 | history | asked | Jim Conant | CC BY-SA 2.5 |