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Equivariant Poincare Duality on G-Manifolds

Equivariant Poincare Duality on G-Manifolds

Autorzy
Wydawnictwo Springer Nature Customer Service Center GmbH
Data wydania 01/05/2021
Wydanie Pierwsze
Liczba stron 348
Forma publikacji książka w miękkiej oprawie
Poziom zaawansowania Dla profesjonalistów, specjalistów i badaczy naukowych
Język angielski
ISBN 9783030704391
Kategorie Podstawy matematyki, Algebra, Grupy i teoria grup, Topologia, Topologia algebraiczna
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Opis książki

This book carefully presents a unified treatment of equivariant Poincare duality in a wide variety of contexts, illuminating an area of mathematics that is often glossed over elsewhere. The approach used here allows the parallel treatment of both equivariant and nonequivariant cases. It also makes it possible to replace the usual field of coefficients for cohomology, the field of real numbers, with any field of arbitrary characteristic, and hence change (equivariant) de Rham cohomology to the usual singular (equivariant) cohomology . The book will be of interest to graduate students and researchers wanting to learn about the equivariant extension of tools familiar from non-equivariant differential geometry.

Equivariant Poincare Duality on G-Manifolds

Spis treści

Preface.- Introduction.- 1 Nonequivariant Background.- 1.1 Category of Cochain Complexes.- 1.2 Some Categories of Manifolds.- 1.3 Poincare Pairing.- 1.4 Manifolds and maps of Finite de Rham Type.- 1.5 Manifolds With Boundary.- 1.6 Proof of Proposition 1.4.4.- 1.7 The Gysin Functor.- 1.8 The Gysin Functor for Proper Maps.- 1.9 Principal Examples of Gysin Morphisms.- 1.10 Constructions of Gysin Morphisms.- 1.11 Exercises.- 1.12 Conclusion.- 2 Equivariant Background.- 2.1 Category of Cochain ggg-Complexes.- 2.2 Equivariant Cohomology of g-Complexes.- 3 Equivariant Cohomology of G-Manifolds.- 3.1 Equivariant Differential Forms.- 3.2 The Borel Construction.- 4 Equivariant Poincare Duality.- 4.1 Differential Graded Modules over a Graded Algebra. 4.2 Deriving Functors.- 4.3 Equivariant Integration.- 4.4 Equivariant Poincare Pairing.- 4.5 G-Equivariant Poincare Duality Theorem.- 4.6 T-Equivariant Poincare Duality Theorem.- 5 Equivariant Gysin Morphism.- 5.1 G-Equivariant Gysin Morphism for General Maps.- 5.2 G-Equivariant Gysin Morphism for Proper Maps.- 5.3 Comparison Theorems.- 5.4 Universal Property of the equivariant Gysin Morphism.- 5.5 Group Restriction.- 5.6 Explicit Constructions of Equivariant Gysin Morphisms.- 5.7 Exercises.- 6 Equivariant Euler Classes.- 6.1 The Nonequivariant Euler Class.- 6.2 G-Equivariant Euler Class.- 6.3 T-Equivariant Euler Classes of a Fixed Point.- 7 Localizations.- 7.1 The Localization Functor.- 7.2 Torsions in Equivariant Cohomology Modules.- 7.3 Localization Theorems.- 8 Changing the coefficients field.- 8.1 Preliminaries.- 8.2 Equivariant Poincare Duality on arbitrary fields.- 8.3 Equivariant Gysin morphisms on arbitrary fields.- 9 Appendix.- Bibliography.- Index.

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