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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.

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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 '11 at 21:41
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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 '11 at 21:58
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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 '11 at 22:00
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@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 '11 at 0:51
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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 '11 at 2:12

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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.)

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This is very close to what I am interested in! –  Jim Conant Feb 12 '11 at 23:17

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