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.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras.

The building blocks of the theory are geometric objects associated to a reflection arrangement such as faces, flats, lunes, and their orbits under the action of the Coxeter group. A generalized notion of zeta and Möbius function play a fundamental role in all aspects of the theory, including exp-log correspondences and results such as the Poincarö–Birkhoff–Witt theorem. The Tits algebra and its invariant subalgebra also play key roles.

This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

We provide a short introduction to the area of Lieb–Thirring inequalities and their applications. We also explain the structure of the book and summarize some of our notation and conventions.