Equivariant Algebraic Topology

This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The works begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. The book then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology. Finally, the book gives an introduction to "brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail.

Filling a gap in the literature, this book takes the reader to the frontiers of equivariant topology, the study of objects with specified symmetries. The discussion is motivated by reference to a list of instructive “toy” examples and calculations in what is a relatively unexplored field. The authors also provide a reading path for the first-time reader less interested in working through sophisticated machinery but still desiring a rigorous understanding of the main concepts. The subject’s classical counterparts, ordinary homology and cohomology, dating back to the work of Henri Poincaré in topology, are calculational and theoretical tools which are important in many parts of mathematics and theoretical physics, particularly in the study of manifolds. Similarly powerful tools have been lacking, however, in the context of equivariant topology. Aimed at advanced graduate students and researchers in algebraic topology and related fields, the book assumes knowledge of basic algebraic topology and group actions.

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.

Intended for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics, this text introduces techniques that are essential in several areas of modern mathematics. With numerous exercises and examples, it covers the core notions and applications of equivariant cohomology.

This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. T.

The first part of this research monograph discusses general properties of G-ENRBs - Euclidean Neighbourhood Retracts over B with action of a compact Lie group G - and their relations with fibrations, continuous submersions, and fibre bundles. It thus addresses equivariant point set topology as well as equivariant homotopy theory. Notable tools are vertical Jaworowski criterion and an equivariant transversality theorem. The second part presents equivariant cohomology theory showing that equivariant fixed point theory is isomorphic to equivariant stable cohomotopy theory. A crucial result is the sum decomposition of the equivariant fixed point index which provides an insight into the structure of the theory's coefficient group. Among the consequences of the sum formula are some Borsuk-Ulam theorems as well as some folklore results on compact Lie-groups. The final section investigates the fixed point index in equivariant K-theory. The book is intended to be a thorough and comprehensive presentation of its subject. The reader should be familiar with the basics of the theory of compact transformation groups. Good knowledge of algebraic topology - both homotopy and homology theory - is assumed. For the advanced reader, the book may serve as a base for further research. The student will be introduced into equivariant fixed point theory; he may find it helpful for further orientation.

A collection of research papers, both new and expository, based on the interests of Professor J. P. C. Greenlees.