Equivariant Cohomology of Configuration Spaces Mod 2

The State of the Art

Pavle V. M. Blagojević, Frederick R. Cohen, Michael C. Crabb, Wolfgang Lück, Günter M. Ziegler (Autoren)

Buch | Softcover

XIX, 210 Seiten

2021 | 1st ed. 2021
Springer International Publishing (Verlag)
978-3-030-84137-9 (ISBN)

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This book gives a brief treatment of the equivariant cohomology of the classical configuration space F( ^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( ^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, Blagojevic, Lück and Ziegler (2016), who used a claimed intermediate result in order to derive lower bounds for the existence of k-regular and -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.

- 1. Snapshots from the History. - Part I Mod 2 Cohomology of Configuration Spaces. - 2. The Ptolemaic Epicycles Embedding. - 3. The Equivariant Cohomology of Pe(Rd, 2m). - 4. Hu'ng's Injectivity Theorem. - Part II Applications to the (Non-)Existence of Regular and Skew
Embeddings. - 5. On Highly Regular Embeddings: Revised. - 6. More Bounds for Highly Regular Embeddings. - Part III Technical Tools. - 7. Operads. - 8. The Dickson Algebra. - 9. The Stiefel-Whitney Classes of the Wreath Square of a Vector Bundle. - 10. Miscellaneous Calculations.

"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)

“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)

Erscheinungsdatum	03.12.2021
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XIX, 210 p. 12 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	358 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	algebraic topology • configuration spaces • Equivariant Cohomology • k-Regular Embeddings • Mathematics
ISBN-10	3-030-84137-5 / 3030841375
ISBN-13	978-3-030-84137-9 / 9783030841379
Zustand	Neuware