In the setting of a proper, cocompact action by a locally compact, unimodular group
on a Riemannian manifold, we construct equivariant spectral
flow of paths of Dirac-type operators. This takes values in the
-theory of the
group
-algebra
of
. In the
case where
is the fundamental group of a compact manifold, the summation map maps equivariant
spectral flow on the universal cover to classical spectral flow on the base manifold.
We obtain “index equals spectral flow” results. In the setting of a smooth path of
-invariant Riemannian
metrics on a
-spin
manifold, we show that the equivariant spectral flow of the
corresponding path of spin Dirac operators relates delocalised
-invariants and
-invariants
for different positive scalar curvature metrics to each other.
Keywords
spectral flow, proper group action, $K$-theory,
$C^*$-algebra