Large scale index of multi-partitioned manifolds

TL;DR

This paper introduces a new multi-partitioned index for spin manifolds with hypersurface partitions, linking scalar curvature bounds to index vanishing and providing a localization technique applicable to twisted Dirac operators.

Contribution

It defines a coarse multi-partitioned index for spin Dirac operators on manifolds partitioned by hypersurfaces and proves a localization property that simplifies index calculations.

Findings

Multi-partitioned index equals the Fredholm index on the intersection manifold N.

Positive scalar curvature bounds imply the multi-partitioned index vanishes.

Localization property holds even with arbitrary Hilbert A-module bundle twisting.

Abstract

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.