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. I thank also my second advisor Ralf Meyer, whom I bothered many times with problems and who patiently and generously answered my questions.I will not lose this opportunity to thank Victor Pidstrygach, whose many lectures and seminars made me more enthusiastic about mathematics. I will also not forget the discussions about life and mathematics in his tea seminars. His lectures in my first semester were one of my main reasons to change to mathematics.My thanks go also to Simone Cecchini, Thorsten Hertl, Christoff Krüger and Vito Felice Zenobi for many memorable mathematical and nonmathematical discussions. Vito's comment on an earlier draft of Chapter 3 improved the presentation of that chapter.The burden of writing this dissertation would not be tolerable without the help of my parents, my brothers Behzad, Hadi and Mehdi and my dear friends Esther Klingenberg, Hanna Kaldenbach and Pablo Schmelzer, who also had to deal with me in my less agreeable moments and with whom I shared (and continue to share) many hilarious moments.Roughly speaking, an operator is called a finite propagation operator if it does not move the support of sections too much. An operator is called locally compact, if after cutting it down to compact regions one obtains compact operators. If the Γ-action on X is cocompact one has the followingSetting X to be the universal cover of a compact spin manifold, we thus obtain the right hand side of the index map using the language of coarse geometry. Now we discuss how to find a model for K-homology using Roe algebras.Definition 1.0...