Concordances in Positive Scalar Curvature and Index Theory

Abstract

We apply the strategy to study of diffeomorphisms via block diffeomorphisms to the world of positive scalar curvature (psc) metrics. For each closed psc manifold $M$, we construct the cubical set $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$ of all psc block metrics, which only encodes concordance information of psc metrics within its homotopy type. We show that $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$ is a cubical Kan set, give a geometric description for the group structure of the combinatorial homotopy groups, and construct a comparison map from the cubical model $\mathcal{R}^+_{\bullet}(M)$ of the space of psc metrics on $M$ to $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$. Next, we build a concordance-themed model for real $K$-theory based on the notion of invertible block Dirac operators and use it to factor the index difference through $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$. In the final part of this thesis, we construct the psc Hatcher spectral sequence, which is a non-index-theoretic tool to get information about the difference of $\mathcal{R}^+_{\bullet}(M)$ and $\widetilde{\mathcal{R}^{+}_{\bullet}}(M)$.