Overview
- Makes a long sequence of papers considerably more accessible
- Gives a corrected new proof of an influential result of Hung
- Contains applications to the existence of k-regular maps
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2282)
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About this book
This book gives a brief treatment of the equivariant cohomology of the classical configuration space F(ℝ^d,n) from its beginnings to recent developments. This subject has been studied intensively, starting with the classical papers of Artin (1925/1947) on the theory of braids, and progressing through the work of Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). The focus of this book is on the mod 2 equivariant cohomology algebras of F(ℝ^d,n), whose additive structure was described by Cohen (1976) and whose algebra structure was studied in an influential paper by Hung (1990). A detailed new proof of Hung's main theorem is given, however it is shown that some of the arguments given by him on the way to his result are incorrect, as are some of the intermediate results in his paper.
This invalidates a paper by three of the authors, Blagojević, Lück and Ziegler (2016), who used a claimed intermediate result in order to derive lower bounds for the existence of k-regular and ℓ-skew embeddings. Using the new proof of Hung's main theorem, new lower bounds for the existence of highly regular embeddings are obtained: Some of them agree with the previously claimed bounds, some are weaker.
Assuming only a standard graduate background in algebraic topology, this book carefully guides the reader on the way into the subject. It is aimed at graduate students and researchers interested in the development of algebraic topology in its applications in geometry.
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Keywords
Table of contents (10 chapters)
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Front Matter
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Mod 2 Cohomology of Configuration Spaces
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Front Matter
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Applications to the (Non-)Existence of Regular and Skew Embeddings
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Front Matter
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Technical Tools
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Front Matter
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Back Matter
Reviews
“The book is well written. … The book will be important for those who study the cohomology rings of configuration spaces.” (Shintarô Kuroki, Mathematical Reviews, November, 2022)
Authors and Affiliations
Bibliographic Information
Book Title: Equivariant Cohomology of Configuration Spaces Mod 2
Book Subtitle: The State of the Art
Authors: Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-030-84138-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Softcover ISBN: 978-3-030-84137-9Published: 02 December 2021
eBook ISBN: 978-3-030-84138-6Published: 01 January 2022
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XIX, 210
Number of Illustrations: 12 b/w illustrations
Topics: Algebraic Topology, Topology