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 $C^{\ast }$-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.