We raise the ante, by explaining why the heuristic in Chapter 1 prima facie fails. The explanation requires some surgery theory, which will be important throughout the remainder of the book, and facts about lattices, from a variety of sources.

The Borel heuristic makes some preictions about group actions, but in this chapter we see that some are reasonably well founded, with some counterexamples at the prime 2, and others are false. Partly this is explained by means of the difference between equivariant and isovariant topology, and also in restricting the class of equivariantly aspherical spaces that are expected to be rigid.

The book closes with a brief survey of some of the techniques that have been used to prove the Farrell–Jones conjecture and the Baum–Connes conjecture. We lightly touch on the role of expanders as an obstruction to some of the proof techniques, and the utility of amenability, and its variant, coarse embedding in Hilbert space, as a tool for proving such conjectures.

This chapter sets the stage for the book, and describes a heuristic connecting rigidity in the Riemannian geometry setting to topological problems.

We discuss the classical construction of locally symmetric manifolds, and also subsequent constructions, due to Davis, Gromov, Piatetski-Shapiro, and Thurston, of aspherical manifolds, to which the Borel conjecture applies.

The Novikov conjecture is put into a larger context, and we explain when the conjectures one is led to in this way tend to be deep facts about the fundamental group, and when they are theorems. We give examples from differential geometry, algebraic geometry, and transformation groups.

Although the Borel conjecture is a uniqueness question, it has implications for problems of whether certain manifolds exist. We shall discuss a conjecture of Wall, and some of its evidence. And also within transformation groups, the Nielsen problem and an analogue of Connes and Raymond. Further afield, we discuss manifolds with excessive symmetry.

The Novikov conjecture arises as a piece of the Borel conjecture, although generalized beyond the setting of aspherical manifolds. This chapted gives several methods, from spltting theorems to index theory to bounded topology, for verifying this conjecture.