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
[< i> Conjecture de Novikov et fibrés presque
plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (1990), 273–277].
Using a natural construction of
[B. Hanke and T. Schick, < i> Enlargeability and index theory< /i> , 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, < i> Distance function, Linear quasi-connections and Chern character< /i> ,
IHES/M/04/27].< /p>

< p> Connes, Gromov and Moscovici [< i> Conjecture de Novikov et fibrés presque
plats< /i> , C. R. Acad. Sci. Paris Sér. I < b> 310< /b> (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.