## Topological Methods in Nonlinear Analysis

### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal. Volume 26, Number 1 (2005), 75-87.

Dates
First available in Project Euclid: 23 June 2016

https://projecteuclid.org/euclid.tmna/1466705431

Mathematical Reviews number (MathSciNet)
MR2179351

Zentralblatt MATH identifier
1093.19005

#### Citation

Mishchenko, Alexander S.; Teleman, Nicolae. Almost flat bundles and almost flat structures. Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 75--87.https://projecteuclid.org/euclid.tmna/1466705431

#### References

• B. Hanke and T. Schick, Enlargeability and index theory , preprint, March 17, 2004 \ref\key 2
• A. Connes, M. Gromov and H. Moscovici, Conjecture de Novikov et fibrés presque plats , C. R. Acad. Sci. Paris Sér. I, 310 (1990), 273–277 \ref\key 3
• N. Teleman, Distance function, Linear quasi-connections and Chern character , IHES/ M/04/27, 11 p