  The main result of this paper is a new and direct proof of the natural
transformation from the surgery exact sequence in topology to the analytic
K-theory sequence of Higson and Roe.
  Our approach makes crucial use of analytic properties and new index theorems
for the signature operator on Galois coverings with boundary. These are of
independent interest and form the second main theme of the paper. The main
technical novelty is the use of large scale index theory for Dirac type
operators that are perturbed by lower order operators.
