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.

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.) 

