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Large scale index of multi-partitioned manifolds

Schick, T ; Sharif University of Technology | 2018

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  1. Type of Document: Article
  2. DOI: 10.4171/JNCG/282
  3. Publisher: European Mathematical Society Publishing House , 2018
  4. Abstract:
  5. 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 on M with connection. 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 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, the multi-partitioned index 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 N with Euclidean space of dimension q. For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result. Mathematics Subject Classification (2010). 19K56, 58J20, 46L80, 58J22. © European Mathematical Society
  6. Keywords:
  7. Coarse geometry ; Coarse index theory ; Hilbert A-module bundles ; Index theory ; K-theory of C∗-algebras ; Large scale geometry ; Large scale index theory ; Obstruction to positive scalar curvature ; Partitioned manifold index theorem ; Positive scalar curvature
  8. Source: Journal of Noncommutative Geometry ; Volume 12, Issue 2 , July , 2018 , Pages 439-456 ; 16616952 (ISSN)
  9. URL: https://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=12&iss=2&rank=2