Rho-classes, index theory and Stolz' positive scalar curvature sequence

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group Gamma, Stephan Stolz introduced the positive scalar curvature exact sequence. Higson and Roe introduced a K-theory exact sequence -> K*(B Gamma) ->(alpha) K-*(C-Gamma*) ->(j) K-*+ 1(D-Gamma*) -> in coarse geometry. The K-theory groups in question are the home of interesting (secondary) invariants, in particular the rho-class rho(Gamma)(g) is an element of K-*(D-Gamma*) of a metric of positive scalar curvature. One of our main results is the construction of a map from the Stolz exact sequence to the Higson-Roe exact sequence (commuting with all arrows), using coarse index theory throughout. The main tool is an index theorem of Atiyah-Patodi-Singer (APS) type. Here, assume that Y is a compact spin manifold with boundary, with a Riemannian metric g which is of positive scalar curvature when restricted to the boundary (and pi(1)(Y) = Gamma). One constructs an APS-index Ind(Gamma)(Y) is an element of K-*(C-Gamma*). This can be pushed forward to j(*)(Ind(Gamma)(Y)) is an element of K-*(D-Gamma*) (corresponding to the 'delocalized part' of the index). The delocalized APS-index theorem then states that j(*)(Ind(Gamma)(Z)) = rho(Gamma)(g(partial derivative Z)) is an element of K-*(D-Gamma*). As a companion to this, we prove a secondary partitioned manifold index theorem for rho-classes.