  In this paper we prove the existence of a natural mapping from the surgery
exact sequence for topological manifolds to the analytic surgery exact sequence
of N. Higson and J. Roe.
  This generalizes the fundamental result of Higson and Roe, but in the
treatment given by Piazza and Schick, from smooth manifolds to topological
manifolds. Crucial to our treatment is the Lipschitz signature operator of
Teleman.
  We also give a generalization to the equivariant setting of the product
defined by Siegel in his Ph.D. thesis. Geometric applications are given to
stability results for rho classes. We also obtain a proof of the
  APS delocalised index theorem on odd dimensional manifolds, both for the spin
Dirac operator and the signature operator, thus extending to odd dimensions the
results of Piazza and Schick.
  Consequently, we are able to discuss the mapping of the surgery sequence in
all dimensions.
