Equivariant algebraic topology pdf

This paper develops the introductory theory of equivariant al gebraic topology. As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic khomology. Can do all of algebraic topology of gspaces with gcategories a gposet partially ordered set is a gcategory with at most one morphism, denoted x y, between any two objects. Glimpses of equivariant algebraic topology department of. Specializing to the universal elliptic curve we obtain in particular equivariant spectra of topological modular forms. Ag following ideas of lurie, we give in this article a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Adams, prerequisites on equivariant stable homotopy for carlssons lecture. In studying topological spaces, one often considers continuous maps. Other cohomology theories see also 19d55, 46l80, 58j20, 58j22 46m20. Pdf we define and study an equivariant version of farbers topological complexity for spaces with a given compact group action. Using its equivalence with cellular cohomology on the category of cw spaces, its graded structure tells me in which dimensions cells live, and the torsion, cup products and steenrod. Introductory lectures on equivariant cohomology princeton.

Methods of algebraic topology cohomology, sheaf and bundle theory, etc. This repository holds lecture notes for andrew blumbergs class on equivariant homotopy theory at ut austin in spring 2017. Motivating examples come from the work of hopkins and his collaborators on algebraic geometry in algebraic topology. Since motivic homotopy theory has also grappled with understanding commutative ring spectra in algebraic geometry, sharing of examples and. Equivariant cohomology and equivariant intersection theory. Following ideas of lurie, we give in this article a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Equivariant algebraic ktheory northeastern university. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant. Equivariant derived algebraic geometry june to june 17, 2016 at the american. Equivariant cohomology in algebraic geometry william fulton eilenberg lectures, columbia university, spring 2007. Our primary reference is the book of chrissginzburg 1, chapters 5 and 6. The proof of the theorem in the category of pathconnected gspaces is given in chapter 4 and follows the proof of the non equivariant case given. Out motivation will be to provide a proof of the classical weyl character formula using a localization result.

Masters thesis, memorial university of newfoundland. Equivariant algebraic geometry stanford university. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems. Equivariant algebraic topology has developed in fits and starts ever since. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. Equivariant cohomology in algebraic geometry 5 example 2.

Can do all algebraic topology of gspaces with gposets. It explains the main ideas behind some of the most striking recent advances in the subject. Equivariant algebraic topology concerns the study of algebraic invariants of spaces with group actions. Browse other questions tagged algebraic topology homologycohomology equivariant cohomology or ask your own question. The goal of these lectures is to give an introduction to equivariant algebraic ktheory.

Given four ordered points on p1, there exists a unique automorphism sending the. Gunnar carlsson, a survey of equivariant stable homotopy theory,topology, vol 31, no. Classically equivariant cohomology is defined as in wikipedia. Introduction to equivariant cohomology in algebraic geometry. Pdf algebraic cycles and equivariant cohomology theories. Since algebraic topology is still developing rapidly any attempt to cover the whole. The second aspect of algebraic topology, homotopy theory, begins again with. We highlight some of the major contributions to algebraic topology in india since the dawn of the 21st century, classified broadly under three heads, namely, manifolds and cell complexes, equivariant topology and deformation theory. This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. Hoskins, philip joseph 1992 equivariant algebraic topology and the equivariant brown representability theorem. April 30, 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. Equivariant coarse homotopy theory and coarse algebraic k.

Equivariant derived algebraic geometry american inst. The rst two chapters introduce the basic structural. Aug 20, 2019 we highlight some of the major contributions to algebraic topology in india since the dawn of the 21st century, classified broadly under three heads, namely, manifolds and cell complexes, equivariant topology and deformation theory. Browse other questions tagged algebraictopology homologycohomology equivariantcohomology or ask your own question. This course will cover some basics of group actions in a variety of settings, for. C cohx, the category of coherent sheaves on an algebraic variety x.

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. The equivariant characteristic classes are viewed as classes in the periodic cyclic cohomology of the crossed product by using. The proof of the theorem in the category of pathconnected gspaces is given in chapter 4 and follows the proof of the nonequivariant case given. We first define gcw complexes and prove some basic. In the monograph equivariant stable homotopy theory, lewis, may, and steinberger cite a monograph the homotopical foundations of algebraic topology by peter may, as in preparation. First defined in the 1950s, it has been introduced into ktheory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. Equivariant algebraic geometry january 30, 2015 note. Equivariant topology of real hyperplane arrangements. Coming from non equivariant homotopy theory, i have come to recognise ordinary cohomology as a powerful tool for probing the topological structure of spaces.

Introduction to equivariant cohomology in algebraic geometry dave anderson. Let x be a topological space equipped with an action of a group g. Did peter mays the homotopical foundations of algebraic. Mona merling university of chicago equivariant algebraic ktheory january 2014 4 25. Equivariant algebraic topology applied to some problems in. An important application of equivariant coarse homotopy theory is in the study of assembly maps which appear in isomorphism conjectures of farrelljones or baumconnes type. Ybe a smooth proper morphism of smooth schemes these hypotheses are not optimal. The combinatorial problems are related to known problems as the. Introduction to equivariant cohomology in algebraic geometry dave anderson april 30, 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. This is a little vague so far, so lets try to digest what it means. Algebraic topology is like this in general, but its particularly acute here.

The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the sullivan conjecture that emphasizes its. This is called algebraic ktheory if we wish to generalize this. Around 1960, pierre conner conjectured that if a compact lie group g acts on a space x, then under certain finiteness conditions the vanishing of the cohomology of x implies the vanishing of the cohomology of the orbit space xg. Office hours will be mondays, 12pm, and by appointment before class on wednesdays. We compute the fixed points of these spectra for the circle group and more generally for tori. Handbook of algebraic topology school of mathematics. Sections 1 and 2 are concerned with actions of compact lie groups on topological spaces, especially on symplectic manifolds.

Tammo tom dieck, transformation groups and representation theory, lecture notes in mathematics 766, springer 1979. English pdf migrated pdf a conversion from original format. The proof of the theorem in the category of pathconnected gspaces is given in chapter 4 and follows the proof of the nonequivariant case given in spanier sp, pp. The equivariant algebraic index theorem is a formula expressing the trace on the crossed product algebra of a deformation quantization with a group in terms of a pairing with certain equivariant characteristic classes. In this thesis we present several results on geometric combinatorics whose solution can be achieved by means of results and tools from algebraic topology. Any help by way of pointing out errors, typos, or clarifications would be much appreciated. Gx, the category of gequivariant vector bundles on a topological space x. The main purpose of this thesis is to give a complete proof of the equivariant brown represcentability theorem, in the process developing the equivariant algebraic topology needed in the final proof. In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. This is called algebraic ktheory if we wish to generalize this last example to the equivariant setting, we have to be. I found the following definition in steenrods cohomology operations in the chapter equivariant cohomology. K 0y chtdy ch q y we want to give some example applications.

As examples of equivariant coarse homology theories we discuss equivariant ordinary coarse homology and equivariant coarse algebraic khomology of an additive category. Consider p1with four unordered points up to automorphisms of p1. Mona merling university of chicago equivariant algebraic ktheory january 2014 8 25. X\to y, and while equivariant topology also considers such maps, there is the additional constraint that each map respects symmetry in both its domain and target space. We prove that the topological invariants of hilbra2g are periodic.

Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forgetcontrol maps. In studying topological spaces, one often considers continuous maps f. Joint number theory topology seminar mona merling johns hopkins university equivariant algebraic ktheory abstract. Equivariant algebraic topology applied to some problems in topological combinatorics abstract. Equivariant homotopy and cohomology theory ebok j p. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. Kvect gx is known as equivariant topological ktheory. Anna marie bohmann, kristen mazur, angelica osorno, viktoriya ozornova, kate ponto, and carolyn yarnall.