Large scale index of multi-partitioned manifolds
| Autor: | Schick, Thomas, Zadeh, Mostafa Esfahani |
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| Rok vydání: | 2013 |
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| Zdroj: | J. Noncommut. Geom. 12 (2018), no. 2, 439-456 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4171/JNCG/282 |
| Popis: | Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a Hermitean bundle with connection on M. We define a coarse multi-partitioned index of the spin Dirac operator on M twisted by E. Our main result is the computation of this multi-partitioned index as the Fredholm index of the Dirac operator on the compact manifold N, twisted by the restriction of E to N. We establish the following main application: if the scalar curvature of M is bounded from below by a positive constant everywhere (or even if this happens only on one of the quadrants defined by the partitioning hypersurfaces) then the multi-partitioned index vanishes. Consequently, ind(D_N) is an obstruction to uniformly positive scalar curvature on M. The proof of the multi-partitioned index theorem proceeds in two steps: first we establish a strong new localization property of the multi-partitioned index which is the main novelty of this note. This we establish even when we twist with an arbitrary Hilbert A-module bundle E (for an auxiliary C*-algebra A). This allows to reduce to the case where M is the product of a compact manifold with Euclidean space. For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result. Comment: 14 pages, AMS-latex; v3 is a substantial revision, following many suggestions of the referee, v4 final version to appear in Journal of Non-commutative Geometry; v5 correction of v4, where by accident an outdated version was uploaded |
| Databáze: | arXiv |
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