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 $\widetilde{ \mathcal{ R}^{ +}_{ \bullet}}(M)$.
Categories: math.GT math.AT math.DG math.KT