int_x^(hx)(g^*α-α)

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