When a discrete group acts on a topological space the quotient can be a complicated space with little structure from the classical point of view. There are nevertheless several methods allowing one to define the cohomology of this quotient in a reasonable way. One is purely topological, via the cohomology of the Borel construction. Another is via the cyclic cohomology of the cross-product algebra. We will describe an explicit construction due to A. Connes which relates these two cohomologies and then apply this construction to the irrational rotation algebra.

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