دانلود کتاب Equivariant Poincaré Duality on G-Manifolds: Equivariant Gysin Morphism and Equivariant Euler Classes (Lecture Notes in Mathematics)

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Preface
Acknowledgements
Contents
1 Introduction
1.1 Equivariant deRham Poincaré Duality
1.2 Equivariant deRham Gysin Morphisms
1.3 Adjunction Properties of Gysin Morphisms
1.4 Equivariant Cohomology Viewed as a Relative Cohomology Theory
1.5 Equivariant Poincaré Duality over Arbitrary Fields
1.6 Conditions on the Group G
1.7 Conditions on Topological Spaces
2 Nonequivariant Background
2.1 Category of Cochain Complexes
2.1.1 Fields in Use
2.1.2 Vector Spaces and Pairings
2.1.3 The Category GV(k) of Graded Spaces
2.1.4 The Subcategories of Bounded Graded Spaces
2.1.5 Graded Algebras over Fields
2.1.6 The Category DGV(k) of Differential Graded Vector Spaces
2.1.7 The Shift Functors
2.1.8 The Functors Hom•k(_,_) and (_⊗k _)•
2.1.9 On the Koszul Sign Rule
2.1.10 The Functor Homk•(_,W)
2.1.11 The Duality Functor
2.2 Categories of Manifolds
2.2.1 Manifolds
2.2.2 The Category of Manifolds
2.2.3 The Category of Manifolds and Proper Maps
2.2.4 G-Manifolds
2.3 Orientation and Integration
2.3.1 Orientability
2.3.1.1 Orientations
2.3.2 Integration
2.4 Poincaré Duality
2.4.1 Poincaré Pairing
2.4.2 The Fundamental Class of an Oriented Manifold
2.5 Poincaré Adjunctions
2.5.1 Poincaré Adjoint Pairs
2.5.2 Manifolds and Maps of Finite deRham Type
2.5.3 Ascending Chain Property
2.5.4 Existence of Proper Invariant Functions
2.5.5 Manifolds with Boundary
2.5.6 Proof of Proposition 2.5.3.1
2.6 The Gysin Functor
2.6.1 The Right Poincaré Adjunction Map
2.6.2 The Gysin Morphism
2.6.3 The Image of DM
2.7 The Gysin Functor for Proper Maps
2.8 Constructions of Gysin Morphisms
2.8.1 The Proper Case
2.8.2 The General Case
3 Poincaré Duality Relative to a Base Space
3.1 Fiber Bundles
3.1.1 Terminology
3.1.2 The Categories TopB and ManB
3.1.3 The Relative Point of View
3.1.4 Fiber Product
3.1.5 The Base Change Functor
3.1.6 Fiber Products of Fiber Bundles of Manifolds
3.1.7 Orientable Fiber Bundles
3.1.8 The Categories ManB, FibB and FibBor
3.1.9 Proper Subspaces of a Fiber Bundle
3.1.10 Differential Forms with Proper Supports
3.1.10.1 Poincaré Lemmas for Ωcv(E) and Ω(E)
3.1.10.2 Sheafification of Ωcv(E) and Ω(E) on the Base Space B
3.2 Integration Along Fibers on Fiber Bundles
3.2.1 The Case of Trivial Euclidean Bundles
3.2.1.1 The Case of General Fiber Bundles
3.2.2 Sheafification of Integration Along Fibers
3.2.3 Thom Class of an Oriented Vector Bundle
3.2.3.1 On Tubular Neighborhoods
3.3 Poincaré Duality for Fiber Bundles
3.3.1 Sheafification of the Poincaré Adjunction
3.3.2 Deriving the Sheafified Poincaré Adjunction Functors
3.3.3 The Poincaré Duality Theorem for Fiber Bundles
3.3.4 Poincaré Duality for Fiber Bundles and Base Change
3.4 Poincaré Duality Relative to a Formal Base Space
3.4.1 Formality of Topological Spaces
3.4.2 Poincaré Duality Relative to Classifying Spaces
3.5 Gysin Morphisms for Fiber Bundles
3.5.1 Gysin Morphisms Relative to a Base Space
3.5.2 Gysin Morphisms for Fiber Bundles and Base Change
3.6 Examples of Gysin Morphisms
3.6.1 Adjointness of Gysin Morphism
3.6.2 Constant Map and Locally Trivial Fibrations
3.6.3 Open Embedding
3.6.4 Proper Base Change
3.6.5 Zero Section of a Vector Bundle
3.6.6 Closed Embedding
3.7 Applications
3.7.1 Gysin Long Exact Sequence
3.7.2 Lefschetz Fixed Point Theorem
3.8 Conclusion
4 Equivariant Background
4.1 Significant Dates in Equivariant Cohomology Theory
4.1.1 Cartan\'s ENS Seminar (1950)
4.1.1.1 The Cartan-Weil Morphisms
4.1.1.2 The Cartan Complex
4.1.1.3 Homotopy Quotients and Formality of the Classifying Space
4.1.2 Borel\'s IAS Seminar (1960)
4.1.2.1 The Borel Construction
4.1.3 Atiyah-Segal: Equivariant K-Theory (1968)
4.1.4 Quillen: Equivariant Cohomology (1971)
4.1.5 Hsiang\'s Book (1975)
4.1.6 Atiyah-Bott and Berline-Vergne: Equivariant deRham Cohomology (1980)
4.2 Category of g-Differential Graded Modules
4.2.1 Field in Use
4.2.2 The Category of g-Modules
4.2.3 g-Differential Graded Modules
4.2.4 g-Differential Graded Algebras
4.2.5 Split g-Complexes
4.3 Equivariant Cohomology of g-Complexes
4.3.1 The g-dg-Algebra S(g∨)
4.3.2 Cartan Complexes
4.3.3 Induced Morphisms on Cartan Complexes
4.3.4 Split G-Complexes
4.4 Equivariant Differential Forms
4.4.1 Fields in Use
4.4.2 G-Fundamental Vector Fields
4.4.3 Interior Products and Lie Derivatives
4.4.4 Complexes of Equivariant Differential Forms
4.4.5 On the Connectedness of G
4.4.6 Splitness of Complexes of EquivariantDifferential Forms
4.5 Cohomological Properties of Homotopy Quotients
4.5.1 Local Triviality of G-Spaces
4.5.2 Slices
4.5.3 Existence of Slices
4.6 Constructing Classifying Spaces
4.6.1 The Milnor Construction
4.6.2 Stiefel Manifolds
4.6.3 Convention
4.7 The Borel Construction
4.7.1 The Homotopy Quotient Functor
4.7.2 On the Cohomology of the Homotopy Quotient
4.7.3 Orientability of the Homotopy Quotient
4.8 Equivariant de Rham Comparison Theorems
4.8.1 Question Iso 1
4.8.2 Question Iso 2
4.8.3 Equivariant Cohomology Comparison Theorem
4.9 Cohomology of Classifying Spaces
4.9.1 Canonicity of the Cohomology of Classifying Spaces
4.9.2 Formality of Classifying Spaces
4.10 Local Equivariant Cohomology
4.10.1 The Long Exact Sequence of Local Equivariant Cohomology
5 Equivariant Poincaré Duality
5.1 Differential Graded Modules over a Graded Algebra
5.1.1 Graded Modules and Algebras over Graded Algebras
5.1.2 The Category of ΩG -Graded Modules
5.2 The Category of ΩG -Differential Graded Modules
5.2.1 Definition
5.2.2 The HomΩG• (_,_) and (_⊗ΩG_)• Bifunctors on DGM(ΩG )
5.2.3 The Duality Functor on DGM(ΩG )
5.2.4 The Forgetful Functor
5.2.5 On the Exactness of Hom•(_,_) and (_ ⊗ _)•
5.3 Comparing the Categories C(GM(ΩG )) and DGM(ΩG )
5.3.1 The Tot Functors
5.3.2 The Hom•ΩG (_,_) bifunctor on C(GM(ΩG ))
5.3.3 The (_ ⊗ΩG _)• Bifunctor on C(GM(ΩG))
5.4 Deriving Functors in GM(ΩG )
5.4.1 Augmentations
5.4.2 Simple Complex Associated with a Bicomplex
5.4.2.1 Spectral Sequences Associated with Bicomplexes
5.4.3 The IR HomΩG•(_,_) and (_)⊗ΩGIL(_) Bifunctors on GM(ΩG)
5.4.4 The Ext• and Tor• Bifunctors
5.4.5 The Duality Functor on D(GM(ΩG ))
5.4.6 The Duality Functor on D(DGM(ΩG ))
5.4.7 Spectral Sequences Associated with IR HomΩG•(_,ΩG)
5.5 Equivariant Integration
5.5.1 Definition
5.5.2 Equivariant Integration vs. Integration Along Fibers
5.6 Equivariant Poincaré Duality
5.6.1 The ΩG -Poincaré Pairing
5.6.2 G-Equivariant Poincaré Duality Theorem
5.6.3 Torsion-Freeness, Freeness and Reflexivity
5.6.4 T-Equivariant Poincaré Duality Theorem
6 Equivariant Gysin Morphism and Euler Classes
6.1 G-Equivariant Gysin Morphism
6.1.1 Equivariant Finite deRham Type Coverings
6.1.2 G-Equivariant Gysin Morphism for General Maps
6.1.3 G-Equivariant Gysin Morphism for Proper Maps
6.1.4 Gysin Morphisms through Spectral Sequences
6.2 Group Restriction and Equivariant Gysin Morphisms
6.2.1 Group Restriction and Equivariant Cohomology
6.2.2 Group Restriction and Integration
6.3 Adjointness of Equivariant Gysin Morphisms
6.3.1 Adjointness Property
6.4 Explicit Constructions of Equivariant Gysin Morphisms
6.4.1 Equivariant Open Embedding
6.4.2 Equivariant Constant Map
6.4.3 Equivariant Projection
6.4.4 Equivariant Fiber Bundle
6.4.5 Zero Section of an Equivariant Vector Bundle
6.4.5.1 The Equivariant Thom Class
6.4.6 Equivariant Gysin Long Exact Sequence
6.4.6.1 Exercises
6.5 Equivariant Euler Classes
6.5.1 The Nonequivariant Euler Class
6.5.2 G-Equivariant Euler Class
6.5.3 G-Equivariant Euler Class of Fixed Points
6.5.4 T-Equivariant Euler Class of Fixed Points
7 Localization
7.1 The Localization Functor
7.2 Localized Equivariant Poincaré Duality
7.3 Localized Equivariant Gysin Morphisms
7.4 Torsion in Equivariant Cohomology Modules
7.4.1 Torsion
7.5 Slices and Orbit Types
7.5.1 The General Slice Theorem
7.5.2 Orbit Type of T-Manifolds
7.6 Localized Gysin Morphisms
7.7 The Localization Formula
7.7.1 Inversibility of Euler Classes
8 Changing the Coefficients Field
8.1 Comments about Notations
8.1.1 Preliminaries
8.2 Sheafification of Cartan Models over Arbitrary Fields
8.2.1 Dictionary
8.2.2 Reformulation of The Poincaré Adjunctions
8.3 Equivariant Poincaré Duality over Arbitrary Fields
8.3.1 The Equivariant Duality Theorem
8.4 Formality of I-6muBG over Arbitrary Fields
8.4.1 The Integral Cohomology of G/T
8.5 Equivariant Gysin Morphisms over Arbitrary Fields
8.5.1 Gysin Morphism for General Maps
8.5.2 Gysin Morphism for Proper Maps
8.6 The Localization Formula over Arbitrary Fields
A Basics on Derived Categories
A.1 Categories of Complexes
A.1.1 The Category of Complexes of an Abelian Category
A.1.2 Extending Additive Functors from Ab to C(Ab)
A.1.3 The Mapping Cone
A.1.4 Homotopies
A.1.4.1 Terminology and Notations
A.1.5 The Homotopy Category K(Ab)
A.1.5.1 Triangles and Exact Triangles
A.1.5.2 Triangulated Categories
A.1.6 The Derived Category D(Ab)
A.1.6.1 Multiplicative Collection of Morphisms
A.1.6.2 Universal Property of Localized Categories
A.1.6.3 The Derived Category D(Ab) as Localization of K(Ab)
A.1.6.4 The Morphisms in the Derived Category D(Ab)
A.1.6.5 Factorization of the Cohomology Functor
A.1.7 The Subcategories C*(Ab), K*(Ab) and D*(Ab)
A.2 Deriving Functors
A.2.1 Extending Functors from Ab to D(Ab)
A.2.2 Extending Functors from C(Ab) to K(Ab)
A.2.3 Extending Functors from K(Ab) to D(Ab)
A.2.3.1 Projective and Injective Objects
A.2.4 Acyclic Resolutions
A.2.5 The Duality Functor on D(DGM(ΩG ))
A.2.5.1 IR Hom•ΩG (_,_) in D(GM(ΩG ))
A.2.5.2 The Categories K(DGM(ΩG )) and D(DGM(ΩG ))
A.2.5.3 Extending Duality from D(GM(ΩG )) to D(DGM(ΩG ))
A.3 DG-Modules over DG-Algebras
A.3.1 K-Injective (A,d)-Differential Graded Modules
A.3.2 Formality of DGA\'s
A.3.3 Formality of DGM\'s
B Sheaves of Differential Graded Algebras
B.1 Mild Topological Spaces
B.2 The Sheaf of Functions OX
B.3 Global Lifting of Germs on OX-modules
B.4 OX-Graded Algebras
B.5 Localization Functor for OX-GA\'s
B.5.1 The Isomorphism Hom•A(A,_)≃Γ( Y;_)
B.5.2 Right Adjoint to Hom•A(A,_)
B.5.2.1 Localization Presheaf Functor
B.5.3 The Localization Functor for OX-GA\'s
B.6 Equivalences of Some Derived Functors in DGM(A)
B.6.1 An Equivalence of Categories
B.6.2 Family of Supports
B.6.3 The functor ΓΦ in GM(A( X))
B.7 OX-Differential Graded Algebras
B.7.1 Localization Functor for OX-DGA\'s
B.8 The Localization Functor for OX-DGA\'s
B.9 Equivalences of Derived Functors in DDGM(A,d)
B.9.1 K-Injective Differential Graded Modules
C Cartan\'s Theorem for g-dg-Ideals
D Graded Ring of Fractions
E Hints and Solutions to Exercises
References
Glossary
Index