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.
Author(s): Pavle V. M. Blagojević, Frederick R. Cohen,. Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler
Series: Lecture Notes in Mathematics
Publisher: Springer
Year: 2021
Language: English
Pages: 216
City: Cham
Preface
Notation
Contents
1 Snapshots from the History
1.1 The Braid Group
1.2 The Fundamental Sequence of Fibrations
1.3 Artin's Presentation of Bn and π1(F(R2,n))
1.4 The Cohomology Ring H*(F(R2,n);Z)
1.5 The Cohomology of the Braid Group Bn
1.6 The Cohomology Ring H*(Bn;F2)
1.7 Cohomology of Braid Spaces
1.8 Homology of Unordered Configuration Spaces
Part I Mod 2 Cohomology of Configuration Spaces
2 The Ptolemaic Epicycles Embedding
3 The Equivariant Cohomology of Pe(Rd ,2m)
3.1 Small Values of m
3.2 The Case m=2
3.3 Cohomology of (XX)Z2Sd-1 and (XX)Z2EZ2
3.4 The Induction Step
3.5 The Restriction Homomorphisms – Three Aspects
3.5.1 A Restriction Homomorphism and the Mùi Invariants
3.5.2 A Restriction Homomorphism and the Dickson Invariants
3.5.3 Two Lemmas
4 Hu'ng's Injectivity Theorem
4.1 Critical Points in Hu'ng's Proof of His Injectivity Theorem
4.2 Proof of the Injectivity Theorem
4.2.1 Prerequisites
4.2.2 Proof of the Dual Epimorphism Theorem
4.3 An Unexpected Corollary
4.3.1 Motivation
4.3.2 Corollary
Part II Applications to the (Non-)Existence of Regular and Skew Embeddings
5 On Highly Regular Embeddings: Revised
5.1 k-Regular Embeddings
5.2 -Skew Embeddings
5.3 k-Regular–Skew Embeddings
5.4 Complex Highly Regular Embeddings
6 More Bounds for Highly Regular Embeddings
6.1 Examples of S2m-Representations and Associated Vector Bundles
6.1.1 Examples of S2m-Representations
6.1.2 Associated Vector Bundles
6.2 The Key Lemma and its Consequences
6.3 Additional Bounds for the Existence of Highly Regular Embeddings
6.4 Additional Bounds for the Existence of Complex Highly Regular Embeddings
Part III Technical Tools
7 Operads
7.1 Definition and Basic Example
7.2 O-Space
7.3 Little Cubes Operad
7.4 Cd-Spaces, An Example
7.5 Cd-Spaces, a Free Cd-Space Over X
7.6 Araki–Kudo–Dyer–Lashof Homology Operations
8 The Dickson Algebra
8.1 Rings of Invariants
8.2 The Dickson Invariants as Characteristic Classes
9 The Stiefel–Whitney Classes of the Wreath Square of a Vector Bundle
9.1 The Wreath Square and the (d-1)-Partial Wreath Square of a Vector Bundle
9.2 Cohomology of B(S2ξ)=S2B(ξ)
9.3 The Total Stiefel–Whitney Class of the Wreath Square of a Vector Bundle
10 Miscellaneous Calculations
10.1 Detecting Group Cohomology
10.2 The Image of a Restriction Homomorphism
10.3 Weyl Groups of an Elementary Abelian Group
10.4 Cohomology of the Real Projective Space with Local Coefficients
10.5 Homology of the Real Projective Space with Local Coefficients
References
Index
