Variations on a Theme of Borel
An Essay on the Role of the Fundamental Group in Rigidity

Explains, using examples, the central role of the fundamental group in the geometry, global analysis, and topology of manifolds.

Shmuel Weinberger (Author)

9781107142596, Cambridge University Press

Hardback, published 8 December 2022

351 pages
23.5 x 15.8 x 2.6 cm, 0.7 kg

In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.

1. Introduction
2. Examples of aspherical manifolds
3. First contact – The proper category
4. How can it be true?
5. Playing the Novikov game
6. Equivariant Borel conjecture
7. Existential problems
8. Epilogue – A survey of some techniques
References
Index.

Subject Areas: Topology [PBP], Algebraic geometry [PBMW], Functional analysis & transforms [PBKF], Groups & group theory [PBG], Algebra [PBF]