Overview
- Provides the homotopy theoretic foundations for surgery theory
- Includes a self-contained account of the Hopf invariant in terms of Z_2-equivariant homotopy
- Covers applications of the Hopf invariant to surgery theory, in particular the Double Point Theorem
Part of the book series: Springer Monographs in Mathematics (SMM)
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About this book
Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds.
Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists.
Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, withmany results old and new.Similar content being viewed by others
Keywords
- MSC (2010): 55Q25, 57R42
- geometric Hopf invariant
- manifolds
- doube points of maps
- double point theorem
- algebraic surgery
- difference construction homotopy
- difference construction chain homotopy
- coordinate-free approach to stable homotopy theory
- inner product spaces
- stable homotopy theory
- Z_2 equivariant homotopy
- bordism theory
- surgery obstruction theory
Table of contents (8 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
Bibliographic Information
Book Title: The Geometric Hopf Invariant and Surgery Theory
Authors: Michael Crabb, Andrew Ranicki
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-319-71306-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Hardcover ISBN: 978-3-319-71305-2Published: 06 February 2018
Softcover ISBN: 978-3-319-89061-6Published: 06 June 2019
eBook ISBN: 978-3-319-71306-9Published: 24 January 2018
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XVI, 397
Number of Illustrations: 1 illustrations in colour
Topics: Algebraic Topology, Manifolds and Cell Complexes (incl. Diff.Topology)