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Equivariant Cohomology of Configuration Spaces Mod 2

The State of the Art

  • Book
  • © 2021

Overview

Authors:
  1. Pavle V. M. Blagojević

    1. Institute of Mathematics, Freie Universität Berlin, Berlin, Germany;, Mathematical Institute of Serbian Academy of Sciences and Arts, Belgrade, Serbia

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  2. Frederick R. Cohen

    1. Department of Mathematics, University of Rochester, Rochester, USA

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  3. Michael C. Crabb

    1. Institute of Mathematics, University of Aberdeen, Aberdeen, UK

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  4. Wolfgang Lück

    1. Mathematisches Institut, Universität Bonn, Bonn, Germany

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  5. Günter M. Ziegler

    1. Institut für Mathematik, Freie Universität Berlin, Berlin, Germany

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  • 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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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xix
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  2. Snapshots from the History

    • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
    Pages 1-20

  3. Mod 2 Cohomology of Configuration Spaces

    1. Front Matter

      Pages 21-21
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    2. The Ptolemaic Epicycles Embedding

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 23-31

    3. The Equivariant Cohomology of \({\mathrm {Pe}(\mathbb {R}^{d} ,2^m)}\)

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 33-61

    4. Hu’ng’s Injectivity Theorem

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 63-92

  4. Applications to the (Non-)Existence of Regular and Skew Embeddings

    1. Front Matter

      Pages 93-93
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    2. On Highly Regular Embeddings: Revised

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 95-135

    3. More Bounds for Highly Regular Embeddings

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 137-159

  5. Technical Tools

    1. Front Matter

      Pages 137-137
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    2. Operads

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 163-172

    3. The Dickson Algebra

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 173-177

    4. The Stiefel–Whitney Classes of the Wreath Square of a Vector Bundle

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 179-185

    5. Miscellaneous Calculations

      • Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
      Pages 187-199

  6. Back Matter

    Pages 201-210
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Keywords

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.

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

  • Institute of Mathematics, Freie Universität Berlin, Berlin, Germany;, Mathematical Institute of Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Pavle V. M. Blagojević

  • Department of Mathematics, University of Rochester, Rochester, USA

    Frederick R. Cohen

  • Institute of Mathematics, University of Aberdeen, Aberdeen, UK

    Michael C. Crabb

  • Mathematisches Institut, Universität Bonn, Bonn, Germany

    Wolfgang Lück

  • Institut für Mathematik, Freie Universität Berlin, Berlin, Germany

    Günter M. Ziegler

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

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