Almost flat bundles and almost flat structures (original) (raw)

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [2]. Using a natural construction of [1], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles over arbitrary CW-spaces using quasi-connections [3]. Connes, Gromov and Moscovici [2] showed that for any almost flat bundle α over the manifold M, the index of the signature operator with values in α is a homotopy equivalence invariant of M. From here it follows that a certain integer multiple n of the bundle α comes from the classifying space Bπ 1 (M). The geometric arguments discussed in this paper allow us to show that the bundle α itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace Y ⊂ Bπ 1 (M) trough the classifying mapping.