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

$125.00 (C)

Part of Cambridge Tracts in Mathematics

  • Date Published: December 2022
  • availability: Available
  • format: Hardback
  • isbn: 9781107142596

$ 125.00 (C)

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About the Authors
  • 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.

    • Introduces tools from a variety of fields, useful to students and researchers in topology, geometry, operator theory, and geometric group theory
    • Uses both true and false variations on the conjecture, to gain a deeper understanding of it
    • Makes much more concrete an area where recent work has been expressed very abstractly
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    Product details

    • Date Published: December 2022
    • format: Hardback
    • isbn: 9781107142596
    • length: 351 pages
    • dimensions: 235 x 158 x 26 mm
    • weight: 0.7kg
    • availability: Available
  • Table of 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

  • Author

    Shmuel Weinberger, University of Chicago
    Shmuel Weinberger is Andrew MacLeish Professor of Mathematics at the University of Chicago. His work is on geometry and topology and their applications. To Weinberger, the only thing cooler than discovering some new geometric result (by any method from any area of mathematics) is discovering a hidden geometric side to the seemingly 'ungeometric'. He has written two other books, one on stratified spaces, and the other on the large-scale structure of spaces of Riemannian metrics using tools from logic. An inaugural Fellow of the American Mathematical Society, he is also a Fellow of the American Academy for the Advancement of Science.

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