Almost flat bundles and almost flat structures
The Novikov conjecture asserts that, for \(M\) a closed oriented \(n\)-manifold with \(\pi_1 M= G\), \(f: M \to BG\) the classifying map, and \(x \in H^{n -4i} (M, Q)\), \(\langle f^* x \cup L_i M, [M] \rangle\) is a homotopy invariant. A proof of this conjecture for a large class of groups was given by \textit{A. Connes, M. Gromov} and \textit{H. Moscovici} [C. R. Acad. Sci., Paris, Sér. I 310, 273--277 (1990; Zbl 0693.53007)] using \(K\)-cohomology and the notion of an almost flat bundle on a manifold. The paper under review gives an alternative definition of flat bundle which extends to bundles over \(CW\)-complexes. The main tool used is the notion of a quasi-connection, previously defined by the second author. As an application it is shown that an almost flat bundle, \(\alpha\) over \(M\) comes from an arbitrarily large compact subset of \(B \pi_1(M)\) via the classifying map, \(f\). Previously it was only known that some integer multiple of \(\alpha\) has that property.
scientific article; zbMATH DE number 5001923