Index theory and partitioning by enlargeable hypersurfaces

Abstract

In this paper we state and prove a higher index theorem for an odd-dimensional connected spin Riemannian manifold (M,g) which is partitioned by an oriented closed hypersurface N. This index theorem generalizes a theorem due to N. Higson in the context of Hilbert modules. Then we apply this theorem to prove that if N is area-enlargeable and if there is a smooth map from M into N such that its restriction to N has non-zero degree, then the scalar curvature of g cannot be uniformly positive.