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.

We reformulate the Baum–Connes conjecture with coefficients by introducing a new crossed product functor for $C^{\ast }$-algebras. All confirming examples for the original Baum–Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum–Connes conjecture become confirming examples for our reformulated conjecture.

We study Farrell Nil-groups associated to a finite-order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension zero).