Equivariant Cohomology of Configuration Spaces Mod 2: The State of the Art

Equivariant cohomology of the classical configuration space F(ℝ^d,n) is discussed, with a focus on mod 2 equivariant cohomology algebras. A new proof of Hung's main theorem is provided, invalidating previous results. New lower bounds for the existence of highly regular embeddings are derived, some of which agree with previous claims and some of which are weaker. The book is aimed at graduate students and researchers interested in algebraic topology and its applications in geometry.

Format: Paperback / softback
Length: 210 pages
Publication date: 02 December 2021
Publisher: Springer Nature Switzerland AG

This book provides a comprehensive exploration of the equivariant cohomology of the classical configuration space F(ℝ^d,n) from its early developments to recent advancements. The study of this subject has been an active area of research, with a rich history dating back to the classical papers of Artin (1925/1947) on the theory of braids. This work has been further developed by notable scholars such as Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969).

The primary focus of this book is on the mod 2 equivariant cohomology algebras of F(ℝ^d,n), which were characterized by Cohen (1976) in terms of their additive structure. An influential paper by Hung (1990) delves into the algebraic structure of these algebras. However, it is revealed that certain arguments presented by Hung in his proof are incorrect, as well as some intermediate results in his paper. This invalidity casts doubt on a previous work by Blagojević, Lück, and Ziegler (2016), which utilized a claimed intermediate result to derive lower bounds for the existence of k-regular and ℓ-skew embeddings.

Utilizing the new proof of Hung's main theorem, this book presents fresh lower bounds for the existence of highly regular embeddings. Some of these bounds align with previously claimed results, while others exhibit weaker strength. This book assumes a standard graduate background in algebraic topology and serves as a valuable guide for graduate students and researchers seeking to delve into the development of algebraic topology in its applications to geometry.

Weight: 366g
Dimension: 156 x 234 x 21 (mm)
ISBN-13: 9783030841379
Edition number: 1st ed. 2021