Large scale index of multi-partitioned manifolds

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 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.