Enlargeability and index theory
By Bernhard Hanke, Thomas Schick. 2004
97 citations recorded by the source corpus.
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).
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f9e5a8810d1973d3c5f44d0b013b4656e4d11c1d
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Citations in the corpus
- L 2 -index theorems, KK-theory, and connections (2005, methodology): “Applications of the explicit index formulas to the study of manifolds with metrics of positive scalar curvature are obtained by Hanke and Schick in [ 6 ].”
- The Dirac Spectrum (2009, background): “Gromov-Lawson’s work - should refer to [178] or to [127].”
- Codimension two index obstructions to positive scalar curvature (2014, methodology): “ncept of enlargeability is not used in our paper; it is entirely based on properties of the Rosenberg index. Because enlargeable spin manifolds have non-vanishing Rosenberg index [HS07, Theorem 1.2], [HS06], [HKRS08, Theorem 1.5], this is no loss of generality. Coarse index theory as developed by Ro”