Variations on a theme of Borel : an essay on the role of the fundamental group in rigidity / Shmuel Weinberger.

Format:

Book

Author/Creator:

Weinberger, Shmuel, author.

Series:

Cambridge tracts in mathematics ; 213.

Cambridge tracts in mathematics ; 213

Language:

English

Subjects (All):

Borel, Armand--Influence.

Borel, Armand.

Rigidity (Geometry).

Manifolds (Mathematics).

Three-manifolds (Topology).

Graph theory.

Fundamental groups (Mathematics).

geometry.

Geometry.

Influence (Literary, artistic, etc.).

Physical Description:

xi, 351 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, United Kingdom : Cambridge University Press, 2023.

Summary:

"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."- publishers

Contents:

Introduction. Introduction to geometric rigidity ; The Borel conjecture ; Notes

Examples of aspherical manifolds. Low-dimensional examples ; Constructions of lattices ; Some more exotic aspherical manifolds ; Notes

First Contact: the proper category. Overview ; K\G/Γ and its Large scale geometry ; Surgery ; Strong approximation ; Property (T) ; Cohomology of lattices ; Mixing the ingredients ; Morals ; Notes

How can it be true? Introduction ; The Hirzebruch Signature theorem ; The Novikov conjecture ; First positive results ; Novikov's theorem ; Curvature, rigidity, and controlled topology ; Surgery, revisited ; Controlled topology, revisited ; The principle of descent ; Secondary invariants ; Notes

Playing the Novikov game. Overview ; Anteing up: introduction to index theory ; Playing the games: what happens in the particular cases? ; The Moral ; Playing the Borel game ; Notes

Equivariant Borel Conjecture. Motivation ; Trifles ; h-Cobordisms ; Cappell's UNil groups ; The simplest nontrivial examples ; Generalities about stratfied spaces ;The equivariant Novikov conjecture ; The Farrell-Jones conjecture ; Connection to embedding theory ; Embedding theory ; Notes

Existential problems. Some quesstions ; The 'Wall conjecture' and variants ; The Nielsen problem and the Conner-Raymond conjecture ; Products: on the difference that a group action makes ; Fibering ; Manifolds with excessive symmetry ; Notes

Epilogue: A survey of some techniques

Codimension-1 Methods ; Induction and Control ; Dynamics and foliated control ; Tensor square trick ; The Baum-Connes conjeture ; A-T-menability, uniform embeddability, and expanders.

Notes:

Includes bibliographical references (pages 311-339) and index.

ISBN:

1107142598

9781107142596

OCLC:

1155086019