Almost Flat Bundles and Almost Flat Structures

Using natural construction of ([1]) we present a simple description of almost flat bundles in the sense of Connes, Gromov and Moscovici ([2]). For this we change a notion of almost flat bundle fixing an almost flat structure on the bundle and extend this notion to arbitrary CW–spaces using quasi-connections. In particular we obtain that each almost flat bundle comes roughly speaking from the classifying space of the fundamental group, more exactly from any sufficiently large compact subspace of the classifying space