### Almost flat bundles and almost flat structures

DOI: http://dx.doi.org/10.12775/TMNA.2005.025

#### 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.

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.

#### Keywords

Almost flat bundles; quasi-connection; classifying space; higher signature

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