Index Theory and Positive Scalar Curvature

Abstract

The aim of this dissertation is to use relative higher index theory to study questions of existence and classification of positive scalar curvature metrics on manifolds with boundary. First we prove a theorem relating the higher index of a manifold with boundary endowed with a Riemannian metric which is collared at the boundary and has positive scalar curvature there, to the relative higher index as defined by Chang, Weinberger and Yu. Next, we define relative higher rho-invariants associated to positive scalar curvature metrics on manifolds with boundary, which are collared at boundary. In order to do this, we define variants of Roe and localisation algebras for spaces with cylindrical ends and use this to obtain an analogue of the Higson-Roe analytic surgery sequence for manifolds with boundary. This is followed by a comparison of our definition of the relative index with that of Chang, Weinberger and Yu. The higher rho-invariants can be used to classify positive scalar curvature metrics up to concordance and bordism. In order to show the effectiveness of the machinery developed here, we use it to give a simple proof of the aforementioned statement regarding the relationship of indices defined in the presence of positive scalar curvature at the boundary and the relative higher index. We also devote a few sections to address technical issues regarding maximal Roe and structure algebras and a maximal version of Paschke duality, whose solutions was lacking in the literature.