Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic ...
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This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.Less
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.001.0001
- Subject:
- Mathematics, Geometry / Topology
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a ...
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Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.Less
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0001
- Subject:
- Mathematics, Geometry / Topology
This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable ...
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This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.Less
This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, ...
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This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.Less
This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.
Araceli Bonifant, Misha Lyubich, and Scott Sutherland
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, ...
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John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.Less
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.
