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'm not sure whether this is what you are after, but this paper, Semiinfinite 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.) 

