Introductory Lectures on Equivariant Cohomology
Loring W. Tu
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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.
Coefficient, CW complex, Cramer's rule, Invariant subspace, Principal bundle, Boundary (topology), Cohomology, Exterior algebra, Lie theory, Theorem, Symplectic manifold, Matrix group, Topological group, Linear function, Tangent space, Euler class, Vector space, Lie algebra, Manifold, Hopf fibration, Diffeomorphism, Comparison theorem, Euler characteristic, Lie derivative, Algebraic topology (object), Fundamental group, Submanifold, Unitary group, Curvature form, Differentiable manifold, Subquotient, Raoul Bott, Tangent bundle, Lie group, Quotient group, Fundamental vector field, Topology, Haar measure, Continuous function, Universal bundle, Equivariant K-theory, Group homomorphism, Lie group action, Oscillatory integral, Homotopy group, Subcategory, Identity element, Classifying space, Integral curve, Fixed point (mathematics), De Rham cohomology, Differential geometry, Contractible space, Dual basis, Stationary phase approximation, Inclusion map, K-theory, Algebraic variety, Frame bundle, Diagram (category theory), Exponentiation, Basis (linear algebra), Structure constants, Orthogonal group, Representation theory, Equivariant map, Unit sphere, Principal ideal domain, Neighbourhood (mathematics), Morphism, Symplectic vector space, Equivariant cohomology, Group action, Fiber bundle, Subgroup, Differential form, Quotient space (topology), Exponential function, Weyl group, Algebraic topology, Group theory, Module (mathematics), Singular homology, Cohomology ring, Exponential map (Lie theory), Homotopy, Spectral sequence, Complex projective space, Submersion (mathematics), Exterior derivative, Ring (mathematics), Vector bundle, Mayer–Vietoris sequence, Mathematics, Local diffeomorphism, Linear algebra, Characteristic class, Topological space, Cellular approximation theorem, Algebraic structure