Large scale index of multi-partitioned manifolds

  • Thomas Schick

    Georg-August-Universität Göttingen, Germany
  • Mostafa Esfahani Zadeh

    Sharif University of Technology, Tehran, Iran

Abstract

Let be a complete n-dimensional Riemannian spin manifold, partitioned by two-sided hypersurfaces which have a compact transverse intersection and which in addition satisfy a certain coarse transversality condition. Let be a Hermitean bundle on with connection. We define a coarse multi-partitioned index of the spin Dirac operator on twisted by .

Our main result is the computation of this multi-partitioned index as the Fredholm index of the Dirac operator on the compact manifold , twisted by the restriction of to .

We establish the following main application: if the scalar curvature of 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 .

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 -module bundle (for an auxiliary -algebra ). This allows to reduce to the case where is the product of with Euclidean space of dimension . For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result.

Cite this article

Thomas Schick, Mostafa Esfahani Zadeh, Large scale index of multi-partitioned manifolds. J. Noncommut. Geom. 12 (2018), no. 2, pp. 439–456

DOI 10.4171/JNCG/282