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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   Γ  , Stephan Stolz introduced the positive scalar curvature exact sequence.  Higson and Roe introduced a   K  ‐theory exact sequence    →   K  *    (  B  Γ  )    →  α    K  *    (   C  Γ  *   )    →  j    K   *  +  1(   D  Γ  *   )   →    in coarse geometry. The   K  ‐theory groups in question are the home of interesting (secondary) invariants, in particular the rho‐classρ  Γ    (  g  )   ∈   K  *    (   D  Γ  *   )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 (andπ  1    (  Y  )   =  Γ   ). One constructs an APS‐indexInd  Γ    (  Y  )   ∈   K  *    (   C  Γ  *   )    . This can be pushed forward toj  *    (   Ind  Γ    (  Y  )   )   ∈   K  *    (   D  Γ  *   )(corresponding to the ‘delocalized part’ of the index). The delocalized APS‐index theorem then states thatj  *    (   Ind  Γ    (  Z  )   )   =   ρ  Γ    (   g   ∂  Z    )   ∈   K  *    (   D  Γ  *   )    .  As a companion to this, we prove a secondary partitioned manifold index theorem for   ρ  ‐classes.

Rho‐classes, index theory and Stolz’ positive scalar curvature sequence