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.



Ordinal-indexed homology theory?
I'm not sure whether this is what you are after, but this paper, [Semi-infinite cycles in Floer theory: Viterbo's Theorem](http://arxiv.org/pdf/0911.3714.pdf), 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.)


