Algebraic models in geometry /

Algebraic models in geometry /

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Bibliographic Details
Author / Creator:Félix, Y. (Yves)
Imprint:Oxford ; New York : Oxford University Press, 2008.
Description:xxi, 460 p. : ill. ; 25 cm.
Language:English
Series:Oxford graduate texts in mathematics ; 17
Oxford graduate texts in mathematics ; 17.
Subject:
Homotopy theory.
Geometry, Algebraic.
Geometry, Algebraic.
Homotopy theory.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/7142698
Hidden Bibliographic Details
Other authors / contributors:Oprea, John.
Tanré, Daniel.
ISBN:9780199206513 (rel)
0199206511 (rel)
019920652X (br)
9780199206520 (br)
Notes:Includes bibliographical references (p. [433]-450) and index.
Summary:This text is aimed at both geometers needing the tools of rational homotopy theory to understand and discover new results concerning various geometric subjects, and topologists who require greater breadth of knowledge about geometric applications of the algebra of homotopy theory.
Other form:Online version: Félix, Y. (Yves) Algebraic models in geometry. Oxford ; New York : Oxford University Press, 2008
Table of Contents:
  • Preface
  • 1. Lie groups and homogeneous spaces
  • 1.1. Lie groups
  • 1.2. Lie algebras
  • 1.3. Lie groups and Lie algebras
  • 1.4. Abelian Lie groups
  • 1.5. Classical examples of Lie groups
  • 1.5.1. Subgroups of the real linear group
  • 1.5.2. Subgroups of the complex linear group
  • 1.5.3. Subgroups of the quaternionic linear group
  • 1.6. Invariant forms
  • 1.7. Cohomology of Lie groups
  • 1.8. Simple and semisimple compact connected Lie groups
  • 1.9. Homogeneous spaces
  • 1.10. Principal bundles
  • 1.11. Classifying spaces of Lie groups
  • 1.12. Stiefel and Grassmann manifolds
  • 1.13. The Cartan-Weil model
  • 2. Minimal models
  • 2.1. Commutative differential graded algebras
  • 2.2. Homotopy between morphisms of cdga's
  • 2.3. Models in algebra
  • 2.3.1. Minimal models of cdga's and morphisms
  • 2.3.2. Relative minimal models
  • 2.4. Models of spaces
  • 2.4.1. Real and rational minimal models
  • 2.4.2. Construction of A PL (X)
  • 2.4.3. Examples of minimal models of spaces
  • 2.4.4. Other models for spaces
  • 2.5. Minimal models and homotopy theory
  • 2.5.1. Minimal models and homotopy groups
  • 2.5.2. Relative minimal model of a fibration
  • 2.5.3. The dichotomy theorem
  • 2.5.4. Minimal models and some homotopy constructions
  • 2.6. Realizing minimal cdga's as spaces
  • 2.6.1. Topological realization of a minimal cdga
  • 2.6.2. The cochains on a graded Lie algebra
  • 2.7. Formality
  • 2.7.1. Bigraded model
  • 2.7.2. Obstructions to formality
  • 2.8. Semifree models
  • 3. Manifolds
  • 3.1. Minimal models and manifolds
  • 3.1.1. Sullivan-Barge classification
  • 3.1.2. The rational homotopy groups of a manifold
  • 3.1.3. Poincaré duality models
  • 3.1.4. Formality of manifolds
  • 3.2. Nilmanifolds
  • 3.2.1. Relations with Lie algebras
  • 3.2.2. Relations with principal bundles
  • 3.3. Finite group actions
  • 3.3.1. An equivariant model for ¿-spaces
  • 3.3.2. Weyl group and cohomology of BG
  • 3.4. Biquotients
  • 3.4.1. Definitions and properties
  • 3.4.2. Models of biquotients
  • 3.5. The canonical model of a Riemannian manifold
  • 4. Complex and symplectic manifolds
  • 4.1. Complex and almost complex manifolds
  • 4.1.1. Complex manifolds
  • 4.1.2. Almost complex manifolds
  • 4.1.3. Differential forms on an almost complex manifold
  • 4.1.4. Integrability of almost complex manifolds
  • 4.2. Kähler manifolds
  • 4.2.1. Definitions and properties
  • 4.2.2. Examples: Calabi-Eckmann manifolds
  • 4.2.3. Topology of compact Kähler manifolds
  • 4.3. The Dolbeault model of a complex manifold
  • 4.3.1. Definition and existence
  • 4.3.2. The Dolbeault model of a Kähler manifold
  • 4.3.3. The Borel spectral sequence
  • 4.3.4. The Dolbeault model of Calabi-Eckmann manifolds
  • 4.4. The Frölicher spectral sequence
  • 4.4.1. Definition and properties
  • 4.4.2. Pittie's examples
  • 4.5. Symplectic manifolds
  • 4.5.1. Definition of symplectic manifold
  • 4.5.2. Examples of symplectic manifolds
  • 4.5.3. Symplectic manifolds and the hard Lefschetz property
  • 4.5.4. Symplectic and complex manifolds
  • 4.6. Cohomologically symplectic manifolds
  • 4.6.1. C-symplectic manifolds
  • 4.6.2. Symplectic homogeneous spaces and biquotients
  • 4.6.3. Symplectic fibrations
  • 4.6.4. Symplectic nilmanifolds
  • 4.6.5. Homotopy of nilpotent symplectic manifolds
  • 4.7. Appendix: Complex and symplectic linear algebra
  • 4.7.1. Complex structure on a real vector space
  • 4.7.2. Complexification of a complex structure
  • 4.7.3. Hermitian products
  • 4.7.4. Symplectic linear algebra
  • 4.7.5. Symplectic and complex linear algebra
  • 4.7.6. Generalized complex structure
  • 5. Geodesics
  • 5.1. The closed geodesic problem
  • 5.2. A model for the free loop space
  • 5.3. A solution to the closed geodesic problem
  • 5.4. A-invariant closed geodesics
  • 5.5. Existence of infinitely many A-invariant geodesics
  • 5.6. Gromov's estimate and the growth of closed geodesics
  • 5.7. The topological entropy
  • 5.8. Manifolds whose geodesics are closed
  • 5.9. Bar construction, Hochschild homology and cohomology
  • 6. Curvature
  • 6.1. Introduction: Recollections on curvature
  • 6.2. Grove's question
  • 6.2.1. The Fang-Rong approach
  • 6.2.2. Totaro's approach
  • 6.3. Vampiric vector bundles
  • 6.3.1. The examples of Özaydin and Walschap
  • 6.3.2. The method of Belegradek and Kapovitch
  • 6.4. Final thoughts
  • 6.5. Appendix
  • 7. G-spaces
  • 7.1. Basic definitions and results
  • 7.2. The Borel fibration
  • 7.3. The toral rank
  • 7.3.1. Toral rank for rationally elliptic spaces
  • 7.3.2. Computation of rk 0 (M) with minimal models
  • 7.3.3. The toral rank conjecture
  • 7.3.4. Toral rank and center of ¿ * (¿M) $$$ Q
  • 7.3.5. The TRC for Lie algebras
  • 7.4. The localization theorem
  • 7.4.1. Relations between G-manifold and fixed set
  • 7.4.2. Some examples
  • 7.5. The rational homotopy of a fixed point set component
  • 7.5.1. The rational homotopy groups of a component
  • 7.5.2. Presentation of the Lie algebra L F = ¿ * (¿F) $$$ Q
  • 7.5.3. Z/2Z-Sullivan models
  • 7.6. Hamiltonian actions and bundles
  • 7.6.1. Basic definitions and properties
  • 7.6.2. Hamiltonian and cohomologically free actions
  • 7.6.3. The symplectic toral rank theorem
  • 7.6.4. Some properties of Hamiltonian actions
  • 7.6.5. Hamiltonian bundles
  • 8. Blow-ups and Intersection Products
  • 8.1. The model of the complement of a submanifold
  • 8.1.1. Shriek maps
  • 8.1.2. Algebraic mapping cones
  • 8.1.3. The model for the complement C
  • 8.1.4. Properties of Poincaré duality models
  • 8.1.5. The configuration space of two points in a manifold
  • 8.2. Symplectic blow-ups
  • 8.2.1. Complex blow-ups
  • 8.2.2. Blowing up along a submanifold
  • 8.3. A model for a symplectic blow-up
  • 8.3.1. The basic pullback diagram of PL-forms
  • 8.3.2. An illustrative example
  • 8.3.3. The model for the blow-up
  • 8.3.4. McDuff's example
  • 8.3.5. Effect of the symplectic form on the blow-up
  • 8.3.6. Vanishing of Chern classes for KT
  • 8.4. The Chas-Sullivan loop product on loop space homology
  • 8.4.1. The classical intersection product
  • 8.4.2. The Chas-Sullivan loop product
  • 8.4.3. A rational model for the loop product
  • 8.4.4. Hochschild cohomology and Cohen-Jones theorem
  • 8.4.5. The Chas-Sullivan loop product and closed geodesics
  • 9. A Florilège of geometric applications
  • 9.1. Configuration spaces
  • 9.1.1. The Fadell-Neuwirth fibrations
  • 9.1.2. The rational homotopy of configuration spaces
  • 9.1.3. The configuration spaces F(R n ,k)
  • 9.1.4. The configuration spaces of a projective manifold
  • 9.2. Arrangements
  • 9.2.1. Formality of the complement of a geometric lattice
  • 9.2.2. Rational hyperbolicity of the space M(A)
  • 9.3. Toric topology
  • 9.4. Complex smooth algebraic varieties
  • 9.5. Spaces of sections and Gelfand-Fuchs cohomology
  • 9.5.1. The Haefliger model for spaces of sections
  • 9.5.2. The Bousfield-Peterson-Smith model
  • 9.5.3. Configuration spaces and spaces of sections
  • 9.5.4. Gelfand-Fuchs cohomology
  • 9.6. Iterated integrals
  • 9.6.1. Definition of iterated integrals
  • 9.6.2. The cdga of iterated integrals
  • 9.6.3. Iterated integrals and the double bar construction
  • 9.6.4. Iterated integrals, the Hochschild complex and the free loop space
  • 9.6.5. Formal homology connection and holonomy
  • 9.6.6. A topological application
  • 9.7. Cohomological conjectures
  • 9.7.1. The toral rank conjecture
  • 9.7.2. The Halperin conjecture
  • 9.7.3. The Bott conjecture
  • 9.7.4. The Gromov conjecture on LM
  • 9.7.5. The Lalonde-McDuff question
  • A. De Rham forms
  • A.1. Differential forms
  • A.2. Operators on forms
  • A.3. The de Rham theorem
  • A.4. The Hodge decomposition
  • B. Spectral sequences
  • B.1. What is a spectral sequence?
  • B.2. Spectral sequences in cohomology
  • B.3. Spectral sequences and filtrations
  • B.4. Serre spectral sequence
  • B.5. Zeeman-Moore theorem
  • B.6. An algebraic example: The odd spectral sequence
  • B.7. A particular case: A double complex
  • C. Basic homotopy recollections
  • C.1. n-equivalences and homotopy sets
  • C.2. Homotopy pushouts and pullbacks
  • C.3. Cofibrations and fibrations
  • References
  • Index

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