Enlargeability and index theory

Abstract

Let M be a closed enlargeable spin manifold. We show non-triviality of the universal index obstruction in the K-theory of the maximal $C^{\ast }$-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 $H_n(B π_1(M) ; \Q)$. This answers a question of Burghelea (1983).

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

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