Welcome to MathOverflow

Question

13

2

Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ordinal? The obvious ways of trying to define an $H_\omega(X)$ don't work out very well essentially because $S^\infty$ is contractible. Still, I've always thought it was an interesting question, and I wonder whether anyone has thought of this or knows of any ideas along these lines.

flag	asked Feb 12 2011 at 21:08 Jim Conant 2,935●1●6●27

4

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… – Peter LeFanu Lumsdaine Feb 12 2011 at 21:41

1

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 $\beta < \alpha$. – Marty Feb 12 2011 at 21:58

1

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? – Tilman Feb 12 2011 at 22:00

1

@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. – Qiaochu Yuan Feb 13 2011 at 0:51

2

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). – Tim Perutz Feb 13 2011 at 2:12

show 8 more comments

Kevin Walker · Accepted Answer · 2011-02-12 23:07:03Z UTC

I'm not sure whether this is what you are after, but this paper, Semi-infinite cycles in Floer theory: Viterbo's Theorem, by Max Lipyanskiy, develops a theory of ($\omega$/2+k)-dimensional cycles in an $\omega$-dimensional manifold with a choice of polarization of its tangent space. ($k\in\mathbb{Z}$ above.)

Ordinal-indexed homology theory?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.