Hanke, B. and Schick, T.: Enlargeability and index theory
- Autor: Bernhard Hanke and Thomas Schick
- Titel: Enlargeability and index theory
(dvi) (pdf)
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Preprint, arXiv;
Journal of Differential geometry 74 (2), 293-320 (2006)
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 $C^*$-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).
Thomas Schick