  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 G, Stephan Stolz
introduced the positive scalar curvature exact sequence, in analogy to the
surgery exact sequence in topology. It calculates a structure group of metrics
of positive scalar curvature on M (the object we want to understand) in terms
of spin-bordism of BG and a somewhat mysterious group R(G).
  Higson and Roe introduced a K-theory exact sequence in coarse geometry which
contains the Baum-Connes assembly map, with one crucial term K(D*G) canonically
associated to G. The K-theory groups in question are the home of interesting
index invariants and secondary invariants, in particular the rho-class in
K_*(D*G) of a metric of positive scalar curvature on a spin manifold.
  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.
  Our main tool are two index theorems, which we believe to be of independent
interest. The first is an index theorem of Atiyah-Patodi-Singer 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 with fundamental group G). Because the Dirac operator on the boundary is
invertible, one constructs a delocalized APS-index in K_* (D*G). We then show
that this class equals the rho-class of the boundary.
  The second theorem equates a partitioned manifold rho-class of a positive
scalar curvature metric to the rho-class of the partitioning hypersurface.
