Part of the lecture notes in mathematics book series lnm, volume 34 log in to check access. For an accessible introduction to the more general theory of bredon equivariant. This is a borel style equivariant cohomology theory. Introduction equivariant cohomology for spaces equipped with an action of a xed groupg was developed by bredon 1 and illman 3. It furthermore reveals a close connection between higher limits and modern finite group theory, and gives an easy systematic way to see which subgroups can be omitted from c without changing the higher limits.
By a gcwcomplex we mean a cwcomplex k with a given cellular action of g such that, for every subgroup h of g, the fixed point set kh is a subcomplex of k. A bredon coefficient system for g is a covariant functor from og to the category of abelian groups. X y induces an isomorphism on these groups, if there is a map g. In addition to the borelstyle equivariant theories, there is another class of equivariant theories originally due to bredon see 10, 33 considered so far for compact group actions on topological spaces. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for. Until now, the methods developed for the computation of k theory. The construction of mug will be treated in detail as the preliminaries for 10. Introduction equivariant cohomology theories are often viewed as very di cult to compute. Andrew putman smith theory and bredon homology notes bredon equivariant cohomology theories smith fixed points of periodic transformations smith theory via bredon homology montag, 23. Advanced topics we give many examples in equivariant stable homotopy theory. Pdf bredon cohomology, k theory and k homology of pullbacks. A new description of equivariant cohomology for totally. Introduction x g bredon homology g university of notre dame.
The ordinary ro c graded bredon cohomology of a point. This means that for every point x2egthere is a neighborhood. It is a generalization to topological groups of an equivariant cohomology theory due to bredon 2 for spaces with an action of a. New trends in triangulated categories and their associated. In this paper we compute the bredon equivariant homology of.
Borel equivariant theories from non equivariant spectra. Then we introduce a generalization of bredon cohomology theories, the socalled roggraded cohomology theories, denoted h g, which are graded on the virtual representations of g, instead of the integers. A cohomology theory on or 0 will be called classical equivariant classical cohomolo gy but lassical equivariant cohomology as defined, for example, in steenrod and epstein, cohomology operations if it satisfies the additional dimension axiom. Also, relative cohomology is introduced into sheaf theory. A new description of equivariant cohomology for totally disconnected groups volume 1 issue 3 christian voigt. Classifying spaces and bredon co homology for transitive groupoids. Borelstyle equivariant k theory of xis the completion of the atiyahsegal equivariant k theory of x see 1 and is therefore a coarser invariant of x. Until now, the methods developed for the computation of k theory and khomology groups of extensions. Reading list for equivariant cohomology mathoverflow. In mathematics, equivariant cohomology or borel cohomology is a cohomology theory from. Eilenbergmoore spectral sequence, bredon cohomology, negative algebraic k theory, chrystallographic groups. Graded bredon cohomology of a point mathematics and applications.
The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic. Equivariant cohomology theories, lecture notes in mathematics, springer verlag 1967 with john w. Representing bredon cohomology with local coefficients. Bredon, is a type of equivariant cohomology that is a contravariant functor from the category of gcomplex with equivariant homotopy maps to the category of abelian groups together with the connecting homomorphism satisfying some conditions. In mathematics, equivariant cohomology or borel cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action. Bredon homology or cohomology with coe cients in a mackey functor is the same as ordinary equivariant roggraded homology or cohomology in the sense of 12. Throughout this work we introduce various notions in equivariant homotopy theory, namely the central concepts of gcwcomplex, mackey functors and g spectra. G m, where g is a finite group, m is a discrete zgmodule, and m is the mackey functor associated to m. Unfortunately, there are no textbooks about bredon cohomology either as far as i know. Computations of bredon cohomology groups associated to these cohomology theories give inputs to a spectral sequence of atiyahhirzebruch type dl98, abutting to the relevant equivariant cohomology groups. The borel construction of the equivariant cohomology h g x goes as follows.
Bredon, equivariant cohomology theories, lecture notes math. Using the localization technique for equivariant cohomology theory we prove a conjecture of g. Equivariant adamshilton construction 11 references 15 1. Equivariant homotopy and cohomology theory alaska notes hill, hopkins, ravenelkervaire invariant one paper t. Nov 07, 1994 in this paper we will study the equivariant cohomology theories defined on the category gcw of gcwcomplexes by bredon in 1. However, in 1966, bredon introduced a more general version of equivariant cohomology i know the. We would like to give the cohomology of a gcomplex so that information regarding the action of g is incorporated. Y x, not necessarily g equivariant, so that the compositions f g idx and g f idy by homotopies that are once again not necessarily g equivariant. Roggraded bredon cohomology is the correct equivariant cohomology theory to consider. Gaunce lewis calculated the rozpgraded cohomology of complex projective spaces in the 1980s, and william kronholm calculated the. Throughout this work we introduce various notions in equivariant homotopy theory, namely the central concepts of gcwcomplex, mackey functors and gspectra. Dimension in bredon cohomology connecting repositories. Click get books and find your favorite books in the online library. Ro ggraded equivariant cohomology theory a dissertation.
Given an orbifold, we construct an orthogonal spectrum representing its stable global homotopy type. Signature redacted bredon cohomology models for the higher limits over o and ac which have been infinite 49, 43, 42, 81. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. Next, we will discuss equivariant cohomology theories. On the other hand, with illmans theorem of smooth g. Mackey functors in equivariant homotopy and cohomology theory. In this paper we will study the equivariant cohomology theories defined on the category gcw of gcwcomplexes by bredon in l. Concerning relative cohomology, it should be noted that sheaftheoretic cohomology is usually considered as a single space theory. A map x y between gspaces is a g equivariant map or. It is in some sense a sequel to the authors previous book in this download pdf file glen e.
It furthermore reveals a close connection between higher limits and modern finite group theory, and gives an easy systematic way to see which subgroups can be omitted from c. As already remarked by others, some equivariant cohomology theories are only sensitive to underlying equivalences borel theories, while others are only invariant under genuine equivalences. There are further issues with borel style cohomology theories as far as riemannroch is considered. These notes are an overview of equivariant stable homotopy theory.
The fact that sheaftheoretic cohomology satisfies 1 the homotopy property is proved for general topological spaces. Sep 01, 2015 for a discrete group g, we represent the bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivariant crossed complexes subsequently, we construct a naive parametrized gspectrum, such that the cohomology theory defined by it reduces to the bredon cohomology with local coefficients when restricted to suspension. Pdf equivariant quantum cohomology of homogeneous spaces. Download full equivariant quantum cohomology of homogeneous spaces book or read online anytime anywhere, available in pdf, epub and kindle. Bredon equivariant cohomology of the space computes the ordinary cohomology of the orbit space quotient by the group action with the chosen coe cients. In order to construct such isomorphism, on one hand, we give a new construction of bredon roggraded equivariant cohomology theory from the sheaftheoretic viewpoint. In full generality, this is often true, as many computations which nonequivariantly were completely in the 1950s and 1960s are still out of. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the segal conjecture and generalized tate. It is a generalization to topological groups of an equivariant cohomology theory due to bredon 2 for spaces with an action of a fixed discrete groups that yields a nice obstruction theory also.
We show that for a discrete groupg, the equivariant cohomology of a gspace xwith glocal coe cients mis isomorphic to the bredon illman cohomology of xwith equivariant local coe cients m. The story for spectra is a bit more complicated because there are even more types. Bredon homology and cohomology theories, first defined by bredon 3, have proved to be one of. One proof uses ordinary or bredon equivariant cohomology. Bigraded equivariant cohomology of real quadrics sciencedirect. Create free account to access unlimited books, fast download and ads free. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. Given a g equivariant cohomology theory, the orbit category org is constructed, and an org cohomology theory is associated in section 4. Introduction in this paper, we construct an equivariant cohomology theory h.
Let eg be a cw complex on which gacts from the left freely and discretely. In this note we prove a version of the classical doldthom theorem for the roggraded equivariant homology functors h. As already remarked by others, some equivariant cohomology theories are only sensitive to underlying equivalences borel theories, while others are only invariant. Briefly, the homotopy theory of representation spheres has appeared in recent. Sep 15, 2020 echie equivariant 91520 cohomology lectures. Buy this book isbn 9783540349730 digitally watermarked, drmfree included format.
Ams proceedings of the american mathematical society. On the other hand, with illmans theorem of smooth gtriangulation of. What is called bredon cohomology after bredon 67a, bredon 67a is the flavor of ordinary g g equivariant cohomology which uses the fine equivariant homotopy theory of topological gspaces that by elmendorfs theorem is equivalent to the homotopy theory of. Equivariant quantum cohomology of homogeneous spaces. Pdf ebooks can be used on all reading devices immediate ebook. We will do some elementary calculations of bredon co homology.
A primer november 1, 2018 1 the borel construction let gbe a discrete group and let g x. Summer school on equivariant homotopy theory shanghai center. A genuine equivariant cohomology theory e a is an exact contravariant functor on aspaces, which admits an roggraded extension so that we have coherent suspension isomorphisms. Bredon cohomology, equivariant k theory, equivariant bordism. Nonorientable surfaces in orientable 3manifolds, inventiones mathematicae, 7, 1969, 83110 the cohomology ring structure of a fixed point set, annals of mathematics, 80, 1964, 524537. As the bredon cohomology groups are concentrated in even degrees the atiyahhirzebruch spectral sequence collapses at the e 2 term, and we get theorem 3. The bredon cohomology groups of y respect to the family of. G m for any smooth manifold mequipped with an action of a compact lie group g, which we call the chiral equivariant cohomology of m. For an accessible introduction to the more general theory of bredon equivariant cohomology, see 19.
Equivariant cohomology for spaces equipped with an action of a fixed group g was developed by bredon 1 and illman 3. See at equivariant cohomology idea for more motivation. G e bredon, equivariant cohomology theories, lecture notes in math. In particular, the bredon cohomology of eg x with constant coe cient systems recovers the borel cohomology. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Homotopy group action and equivariant cohomology theories. For that reason, no algebraic topologist today would consider writing a book just about borel cohomology. Bredon theory of modules over a category encyclopedia of.
Introduction in bre67, bredon created equivariant homology and cohomology theories of gspaces, now called bredon homology and bredon cohomology, which yield the usual singular homology and cohomology theories when the group acting is taken to be. Orthogonal spectra now represent orbifold cohomology theories which automatically satisfy certain properties as additivity and the existence of mayervietoris sequences. This has led to results in equivariant obstruction theory, orientation theory, and covering spaces among others 7, 8. Smith theory on xed point spaces the conner conjecture on orbit spaces the oliver transfer and roggraded cohomology mackey functors for nite and compact lie groups extending bredon cohomology to roggrading a glimpse of the modern world of spectra and g. Smith theory and bredon homology university of notre dame. By using these constructions we define equivariant bredon homology and cohomology for actions of transitive topological groupoids.
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