Mod Two Homology and Cohomology

Nonfiction, Science & Nature, Mathematics, Topology, Geometry
Cover of the book Mod Two Homology and Cohomology by Jean-Claude Hausmann, Springer International Publishing
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Jean-Claude Hausmann ISBN: 9783319093543
Publisher: Springer International Publishing Publication: January 8, 2015
Imprint: Springer Language: English
Author: Jean-Claude Hausmann
ISBN: 9783319093543
Publisher: Springer International Publishing
Publication: January 8, 2015
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Springer International Publishing

Cover of the book Remote Observatories for Amateur Astronomers by Jean-Claude Hausmann
Cover of the book Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology by Jean-Claude Hausmann
Cover of the book Modelling and Implementation of Complex Systems by Jean-Claude Hausmann
Cover of the book Geology of Coal Deposits of South Limburg, The Netherlands by Jean-Claude Hausmann
Cover of the book Advances in MALDI and Laser-Induced Soft Ionization Mass Spectrometry by Jean-Claude Hausmann
Cover of the book Fibrous Proteins: Structures and Mechanisms by Jean-Claude Hausmann
Cover of the book Partition and the Practice of Memory by Jean-Claude Hausmann
Cover of the book Emotion, Affect and Personality in Speech by Jean-Claude Hausmann
Cover of the book Medical Management of Psychotropic Side Effects by Jean-Claude Hausmann
Cover of the book Theory of Quantum Transport at Nanoscale by Jean-Claude Hausmann
Cover of the book Open Problems in Optimization and Data Analysis by Jean-Claude Hausmann
Cover of the book Image and Graphics by Jean-Claude Hausmann
Cover of the book Implications of Open Access Repositories Quality Criteria and Features for Teachers’ TPACK Development by Jean-Claude Hausmann
Cover of the book Orthopaedic Trauma in the Austere Environment by Jean-Claude Hausmann
Cover of the book Women of Words in Le Morte Darthur by Jean-Claude Hausmann
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy