Almost flat bundles and almost flat structures

Abstract

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [Conjecture de Novikov et fibrés presque plats, C. R. Acad. Sci. Paris Sér. I 310 (1990), 273–277]. Using a natural construction of [B. Hanke and T. Schick, Enlargeability and index theory, preprint, 2004], 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 [N. Teleman, Distance function, Linear quasi-connections and Chern character, IHES/M/04/27].

Connes, Gromov and Moscovici [Conjecture de Novikov et fibrés presque plats, C. R. Acad. Sci. Paris Sér. I 310 (1990), 273–277] showed that for any almost flat bundle $\alpha$ over the manifold $M$, the index of the signature operator with values in $\alpha$ is a homotopy equivalence invariant of $M$. From here it follows that a certain integer multiple $n$ of the bundle $\alpha$ comes from the classifying space $B\pi_{1}(M)$. The geometric arguments discussed in this paper allow us to show that the bundle $\alpha$ itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace $Y\subset B\pi_{1}(M)$ trough the classifying mapping.