Enlargeability and index theory

Let M be a closed enlargeable spin manifold. We show nontriviality 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 π1(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 → Bπ1(M) sends the fundamental class of M to a nontrivial homology class inHn(Bπ1(M);Q). This answers a question of Burghelea (1983).