Variations on a theme of Borel: an essay on the role of the fundamental group in rigidity
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 cent...
Gespeichert in:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY
Cambridge University Press
2023
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| Schriftenreihe: | Cambridge tracts in mathematics
213 |
| Schlagworte: | |
| Online-Zugang: | DE-12 DE-634 DE-92 Volltext |
| Zusammenfassung: | 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 |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 24 Nov 2022) Introduction -- Mostow -- The Borel conjecture -- Notes -- Examples of aspherical manifolds -- Low dimensional examples -- Constructions of lattices (arithmetic and non-arithmetic) -- Some more exotic aspherical manifolds -- Notes -- First Contact -- Overview -- Large scale geometry of KG/Γ and its boundary -- Surgery -- Strong approximation -- Property (T) -- Appendix: Property (T) and expanders -- Cohomology of lattices -- Completion of the discussion -- 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: Index theory -- Appendix: Through the looking glass -- Playing the games: what happens in the particular cases? -- The Moral -- Playing the Borel game -- Notes -- Equivariant Borel Conjecture -- Motivation via Borel heuristic -- Cappell's UNil and its significance -- Connections to embedding theory (embedded Borel conjecture) and the -- failure of EBC because it isn't IBC -- Tate cohomology and its role in classifying actions -- Appendix: Concordance embedding groups -- The Farrell-Jones conjecture's implications -- Existential problems -- Also potential replacement to CR conjecture -- Wall conjecture and Fowler's theorem -- Nielsen problem and the Conner Raymond conjecture -- Products: Where an action makes a difference -- Fibering theorems, and manifolds with small universal covers -- Manifolds with excessive symmetry -- Epilogue: A survey of some techniques -- Codimension one -- Induction and Control -- K-theory and Dynamics -- Tensor square trick -- Higson-Kasparov, -- Lafforgue -- Skandalis-Tu-Yu, -- Some embeddings: hyperbolic groups, linear groups -- Expanders and the failure of the Baum-Connes conjecture with coef |
| Beschreibung: | 1 Online-Ressource (xi, 351 Seiten) |
| ISBN: | 9781316529645 |
| DOI: | 10.1017/9781316529645 |
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| 520 | |a 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 | ||
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| index_date | 2024-07-03T21:22:01Z |
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| spelling | Weinberger, Shmuel 1963- (DE-588)1078286957 aut Variations on a theme of Borel an essay on the role of the fundamental group in rigidity Shmuel Weinberger, University of Chicago Cambridge, United Kingdom ; New York, NY Cambridge University Press 2023 1 Online-Ressource (xi, 351 Seiten) txt rdacontent c rdamedia cr rdacarrier Cambridge tracts in mathematics 213 Title from publisher's bibliographic system (viewed on 24 Nov 2022) Introduction -- Mostow -- The Borel conjecture -- Notes -- Examples of aspherical manifolds -- Low dimensional examples -- Constructions of lattices (arithmetic and non-arithmetic) -- Some more exotic aspherical manifolds -- Notes -- First Contact -- Overview -- Large scale geometry of KG/Γ and its boundary -- Surgery -- Strong approximation -- Property (T) -- Appendix: Property (T) and expanders -- Cohomology of lattices -- Completion of the discussion -- 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: Index theory -- Appendix: Through the looking glass -- Playing the games: what happens in the particular cases? -- The Moral -- Playing the Borel game -- Notes -- Equivariant Borel Conjecture -- Motivation via Borel heuristic -- Cappell's UNil and its significance -- Connections to embedding theory (embedded Borel conjecture) and the -- failure of EBC because it isn't IBC -- Tate cohomology and its role in classifying actions -- Appendix: Concordance embedding groups -- The Farrell-Jones conjecture's implications -- Existential problems -- Also potential replacement to CR conjecture -- Wall conjecture and Fowler's theorem -- Nielsen problem and the Conner Raymond conjecture -- Products: Where an action makes a difference -- Fibering theorems, and manifolds with small universal covers -- Manifolds with excessive symmetry -- Epilogue: A survey of some techniques -- Codimension one -- Induction and Control -- K-theory and Dynamics -- Tensor square trick -- Higson-Kasparov, -- Lafforgue -- Skandalis-Tu-Yu, -- Some embeddings: hyperbolic groups, linear groups -- Expanders and the failure of the Baum-Connes conjecture with coef 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 Rigidity (Geometry) Manifolds (Mathematics) Three-manifolds (Topology) Graph theory Chirurgie Mathematik (DE-588)4200269-2 gnd rswk-swf Novikov-Vermutung (DE-588)4402781-3 gnd rswk-swf Asphärische Mannigfaltigkeit (DE-588)4242581-5 gnd rswk-swf Starrheit Mathematik (DE-588)4326739-7 gnd rswk-swf Asphärische Mannigfaltigkeit (DE-588)4242581-5 s Chirurgie Mathematik (DE-588)4200269-2 s Novikov-Vermutung (DE-588)4402781-3 s Starrheit Mathematik (DE-588)4326739-7 s DE-604 Erscheint auch als Druck-Ausgabe 978-1-107-14259-6 https://doi.org/10.1017/9781316529645 Verlag URL des Erstveröffentlichers Volltext |
| spellingShingle | Weinberger, Shmuel 1963- Variations on a theme of Borel an essay on the role of the fundamental group in rigidity Rigidity (Geometry) Manifolds (Mathematics) Three-manifolds (Topology) Graph theory Chirurgie Mathematik (DE-588)4200269-2 gnd Novikov-Vermutung (DE-588)4402781-3 gnd Asphärische Mannigfaltigkeit (DE-588)4242581-5 gnd Starrheit Mathematik (DE-588)4326739-7 gnd |
| subject_GND | (DE-588)4200269-2 (DE-588)4402781-3 (DE-588)4242581-5 (DE-588)4326739-7 |
| title | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity |
| title_auth | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity |
| title_exact_search | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity |
| title_exact_search_txtP | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity |
| title_full | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity Shmuel Weinberger, University of Chicago |
| title_fullStr | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity Shmuel Weinberger, University of Chicago |
| title_full_unstemmed | Variations on a theme of Borel an essay on the role of the fundamental group in rigidity Shmuel Weinberger, University of Chicago |
| title_short | Variations on a theme of Borel |
| title_sort | variations on a theme of borel an essay on the role of the fundamental group in rigidity |
| title_sub | an essay on the role of the fundamental group in rigidity |
| topic | Rigidity (Geometry) Manifolds (Mathematics) Three-manifolds (Topology) Graph theory Chirurgie Mathematik (DE-588)4200269-2 gnd Novikov-Vermutung (DE-588)4402781-3 gnd Asphärische Mannigfaltigkeit (DE-588)4242581-5 gnd Starrheit Mathematik (DE-588)4326739-7 gnd |
| topic_facet | Rigidity (Geometry) Manifolds (Mathematics) Three-manifolds (Topology) Graph theory Chirurgie Mathematik Novikov-Vermutung Asphärische Mannigfaltigkeit Starrheit Mathematik |
| url | https://doi.org/10.1017/9781316529645 |
| work_keys_str_mv | AT weinbergershmuel variationsonathemeofborelanessayontheroleofthefundamentalgroupinrigidity |