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Let X be a connected topological space admitting a universal cover. Let a bea degree one cohomology class on X. We define and study a two-cocycle on agroup acting on X by homeomorphisms preserving the class a. We use this cocycleto investigate group actions on X. For example, we show that if an actionpreserves a Borel probability measure on X then the cocycle is cohomologicallytrivial. Under various assumptions on a homeomorphism g, we prove that it isundistorted in Homeo(X,a). In particular, we introduce a local rotation numberof a homeomorphism and prove that a homeomorphism with non-constant localrotation number is undistorted.

\int_x^{hx}(g^*\alpha-\alpha)