Pdf wittenhodge theory for manifolds with boundary and. By yonedas lemma, to give the structure of an equivariant sheaf to an module f is the same as to give group homomorphisms for rings r over. Foundations of an equivariant cohomology theory for banach. This gives rise to a third relative cohomology theory which we call stratifold tate cohomology. Bredon, equivariant cohomology theories, lecture notes in mathematics, vol. The interaction of this torus equivariant map with the bruhat order and its action on line bundles lead to formulas for its pullback on the equivariant cohomology ring and on equivariant k theory. The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology see there for details called bredon cohomology. In addition to these cohomology theories there are other cohomology theories called weil cohomology theories which behave similarly to singular cohomology. Dylan wilson, equivariant k theory, 2016 pdf, pdf the equivariant chern character is discussed in. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.
Andrew blumberg, equivariant homotopy theory, 2017 pdf, github a standard text is peter may, equivariant homotopy and cohomology theory, cbms regional conference series in mathematics, vol. That is, applying a symmetry transformation and then computing the function produces the same result as computing the. At the heart of the theory is the vertex algebra commutant construction, which plays the same role that ordinary invariant theory plays in classical equivariant cohomology. If the action of g is free, then the notion of an equivariant sheaf simplifies to a sheaf on the quotient xg, because of the descent along torsors. We show that equivariant cohomology is poincare dual to backwards homology, and equivariant homology is poincare dual to backwards cohomology. In other words, that the theory is treelevel exact 10. We show that tate cohomology is an obstruction for poincare duality, which means that it holds if and only if tate cohomology vanishes which is the case if and only if the action is free. We wish to construct equivariant chern classes for real bundles in equivariant cohomology groups with integral coe cients and our main requirement is that one recovers the classical chern classes by forgetting the c 2action. Equivariant cohomology with local coefficients dmlcz czech. Lectures on 2d yangmills theory, equivariant cohomology. Equivariant stratifold homology and cohomology theories for. Foundations of an equivariant cohomology theory for banach algebras, ii kjeld knudsen jensen1 university of pennsylvania, philadelphia, philadelphia 19104 received may 2, 1995.
My understanding is that the plan is for these notes to be compiled into a book at some point. It also enlightens the classical faddeev popov gauge fixing procedure. Equivariant cohomology and equivariant intersection theory michel brion this text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by d. For a general group action and if zhas no boundary, then every g equivariant.
Bg0inducing an isomorphism of homotopy groups and compatible with the identi cation of the fundamental group of each with g. Our main result here is an explicit operator description of the equivariant gromovwitten theory of p1. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. Our main aim is to obtain explicit descriptions of. Another useful computational tool is the blowup sequence. Stratifolds and equivariant cohomology theories ulb bonn.
Chiral equivariant cohomology 5 us the chiral equivariant cohomology of m, which we denote by h. Equivariant ktheory and equivariant cohomology request pdf. Wittenhodge theory for manifolds with boundary and equivariant cohomology article pdf available in differential geometry and its applications 302. In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another. Introduction to equivariant cohomology in algebraic geometry. We establish a localization theorem of borelatiyahsegal type for the equivariant operational k theory of anderson and payne ap. We construct such an equivariant cohomology theory, which we denote by dh g x and call the backwards cohomology. Equivariant cohomology theories and the pattern map article pdf available in houston journal of mathematics 432 june 2015 with 53 reads how we measure reads.
Equivariant cohomology theories and the pattern map. Introduction to equivariant cohomology in algebraic geometry dave anderson january 28, 2011 abstract introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. The bordism theory, originally invented by thom, has been generalized to a generalized homology theory by atiyah, and independently by conner and floyd. Kr theory is a variant of k theory for equivariant spaces with an involution or, in other terms, an action of the group z2. This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. German stefanich, chern character in twisted and equivariant k theory.
Part of the lecture notes in mathematics book series lnm, volume 34 log in to check access. The purpose of this thesis is to describe in detail duggers 6 construction of the spectral sequence, which computes kr theory of an equivariant space from its equivariant cohomology. Thus, a rst na ve, but natural, place to look for such classes is in the integral equivariant cohomology h c 2. A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. There is a conjectured theory of motives which underlie all of the weil cohomology theories. In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. Quite some time passed before algebraic geometers picked up on these ideas, but in the last. The final prices may differ from the prices shown due to specifics of vat rules. Pdf equivariant cohomology theories and the pattern map. Exercises in equivariant cohomology and topological theories. Meanwhile, a similar phenomenon occured to the k theory, originally invented by grothendieck, which was responsible for solutions of many delicate problems in differential topology. Reading list for equivariant cohomology mathoverflow.
The equivariant theory of p1 will play a crucial role in the derivation of the virasoro constraints for target curves in 25. Section 4, based on 14, presents edidin and grahams localization the orem concerning equivariant chow groups for torus actions, and its refined version for. Mackey functors, km,ns, and roggraded cohomology 25 6. In studying topological spaces, one often considers continuous maps. A contractible space on which a topological group g acts freely is familiar from homotopy theory as the total space of a universal principal. The moment map and equivariant cohomology sciencedirect. Passing to the universal covers produces an equivariant map eg. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology.
Discussion relating to k theory of homotopy quotientsborel constructions is in. In the second and third sections we introduce torus equivariant ordinary cohomology and torus equivariant k theory, along with some of their properties and holomorphic analogues. This is the kind of target we are used to from the classical constructions of equivariant elliptic cohomology, for example by grojnowski gro07. Equivariant cohomology and equivariant intersection theory. Equivariant cohomology is suggested as an alternative algebraic framework for the definition of topological field theories constructed by e. Kohomologietheorie cohomology fibrations function function space homology. That is, we prove a combinatorial rule for the structure coef. When a compact lie group gacts on a tame strong polyfold bundle p.
Pdf algebraic cycles and equivariant cohomology theories. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of brst. After youve bought this ebook, you can choose to download either the pdf version or the epub, or both. It explains the main ideas behind some of the most striking recent advances in the subject. Equivariant cohomology in the framework of smooth manifolds is the subject of this book which is part of a collection of volumes edited by j. Equivariant cohomology theories lecture notes in mathematics series by glen e. Algebraic cycles and equivariant cohomology theories article pdf available in proceedings of the london mathematical society s3733 november 1996 with reads how we measure reads. In mathematics, equivariant cohomology or borel cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory.
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