Group algebras are indeed twisted by 2-cocycles, but I believe this is a version of the group algebra that lives in C, so the associativity gets twisted by the associator.

Although this is not quite what I was looking for, it is quite interesting. Thanks for answering.

-1. I agree with Tom.

Are we assuming X is chosen from the union of A_i with uniform probability? Even if so, we can only put bounds on the probability, since we don't seem to know the sizes of the intersections between the other sets.

Suspension shifts the dimension of singular chains up by one. One then runs into orientation considerations when gluing (as in Eric's question), and it changes their parity when looking at products.

Also, Gaussian Blur is a convolution filter on some image manipulation programs that is often used to test computer speed.

I'm afraid cyclic groups are deceptively easy - one can find the Morita invariance classes with just a quadratic form calculation, with no visible topology. This also applies to a certain class of cocycles of abelian groups (Mason, Ng, math.QA/0002246). BG tends to get rather complicated at about the same level of complexity where these techniques fail.

I'm mostly looking for something that works. Since vector spaces don't detect extensions by BH, I guess it's categories or something with similar complexity.

The Atiyah-Singer index theorem also holds for manifolds with no complex structure. Could you explain why it is relevant?

@Ben, I like your interpretation of the question, but I disagree with your implicit claim that the question was even well-defined as stated.