Authors:
- Presents a simplified version of these important tools of algebraic topology
- Provides a self-contained introduction to mod 2 (co)homology
- Begins with basic principles and leads up to advanced topics that are not usually mentioned in an algebraic topology text
- Includes many interesting applications of mod 2 homology and cohomology theory, both classical and more recent
- Includes supplementary material: sn.pub/extras
Part of the book series: Universitext (UTX)
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Table of contents (11 chapters)
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Front Matter
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Back Matter
About this book
Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages:
1. It leads more quickly to the essentials of the subject,
2. An absence of signs and orientation considerations simplifies the theory,
3. Computations and advanced applications can be presented at an earlier stage,
4. Simple geometrical interpretations of (co)chains.
Mod 2 (co)homology was developed in the first quarter of the twentieth century as an alternative to integral homology, before both became particular cases of (co)homology with arbitrary coefficients.
The first chapters of this book may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and singular mod 2 (co)homology are introduced, with their products and Steenrod squares, as well as equivariant cohomology. Classical applications include Brouwer's fixed point theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith theory, Kervaire invariant, etc. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus. More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. Each chapter ends with exercises, with some hints and answers at the end of the book.
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Authors and Affiliations
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University of Geneva, Geneva, Switzerland
Jean-Claude Hausmann
About the author
Jean-Claude Hausmann is honorary professor of Mathematics at the University of Geneva. His domains of interest include topology of high-dimensional manifolds, with applications to configuration spaces inspired by topological robotics and Hamiltonian geometry.
Bibliographic Information
Book Title: Mod Two Homology and Cohomology
Authors: Jean-Claude Hausmann
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-319-09354-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2014
Softcover ISBN: 978-3-319-09353-6Published: 20 January 2015
eBook ISBN: 978-3-319-09354-3Published: 08 January 2015
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: IX, 535
Number of Illustrations: 9 b/w illustrations
Topics: Algebraic Topology, Manifolds and Cell Complexes (incl. Diff.Topology)