Variations on a Theme of Borel : An Essay on the Role of the Fundamental Group in Rigidity (Cambridge Tracts in Mathematics)

Weinberger, Shmuel

Cambridge University Press（2022/12発売）

• 製本 Hardcover:ハードカバー版／ページ数 364 p.
• 言語 ENG
• 商品コード 9781107142596
• DDC分類 516

Full Description

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.

Contents

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.