Enlargeability and index theory

Details Enlargeability and index theory Enlargeability and index theory

2004-03-16

arxiv.org/abs/math/0403257v4

Journal of Differential geometry 74 (2), 293-320 (2006)

Mathematics

Geometric Topology

32 pages, final version accepted for publication, added relation to Gromov's 1-systole, typos corrected; to appear in Journal

Scientific paper

Let M be a closed enlargeable spin manifold. We show non-triviality of the universal index obstruction in the K-theory of the maximal $C^*$-algebra of the fundamental group of M. Our proof is independent from the injectivity of the Baum-Connes assembly map for the fundamental group of M and relies on the construction of a certain infinite dimensional flat vector bundle out of a sequence of finite dimensional vector bundles on M whose curvatures tend to zero. Besides the well known fact that M does not carry a metric with positive scalar curvature, our results imply that the classifying map $M \to B \pi_1(M)$ sends the fundamental class of M to a nontrivial homology class in $H_n(B \pi_1(M) ; \Q)$. This answers a question of Burghelea (1983).

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