I the homotopy limit problem for karoubis hermitian ktheory 23 was posed by thomason in 1983 43. Purity theorem in motivic homotopy theory mit mathematics. Isaksen find an algebraic model for rational motivic homotopy theory. The canonical homomorphisms of topological g spaces are g equivariant continuous functions, and the canonical choice. The new model structure introduced here samples a comparison to the one by voevodsky and hukrizormsby. Various important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a speci.
E ect of a1localization on ring and module structures and on modlcompletions 5. In these notes i will talk about the purity theorem in motivic homotopy theory. Equivariant cycles and cancellation for motivic cohomology j. These notes explore equivariant homotopy theory from the perspective of model categories. We must therefore spend quite some time establishing analogues of. In the process of my research, i realized it would be possible to construct motivic analogues of unoriented and oriented cobordism, which i refer to as mglo and mslo respectively. This includes the naive gspectra which constitute the actual stabilization of equivariant homotopy theory, but is more general, one speaks of genuine g g spectra. This thesis makes progress in computing the coefficients of algebraic hermitian cobordism mglr, a motivic z2equivariant spectrum constructed by p. The above constructions may be unified to apply for all groups at once, this is the content of global equivariant homotopy theory in more general model categories. Equivariant stable homotopy theory over some topological group gg is the stable homotopy theory of gspectra. We show that it allows to detect equivariant motivic weak equivalences on fixed. He was motivated by his work with atiyah 9 on equivariant ktheory, generalizing an. We show that it allows to detect equivariant motivic weak equivalences on fixed points and how this property leads to a.
Workshop on motivic and equivariant homotopy theory october 4th 7th, 2017. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. Karoubis hermitian ktheory can be shown to be a z2equivariant motivic spectrum in our sense. This repository holds lecture notes for andrew blumbergs class on equivariant homotopy theory at ut austin in spring 2017. N2 for a finite galois extension of fields lk with galois group g, we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence. The foundations of equivariant stable motivic homotopy theory 1.
The main reference for this theory is the ams memoir 16 by mandell and may. Galois equivariance and stable motivic homotopy theory. This is a conference on homotopy theory and related geometry and category theory to be held at. Thec 2equivariant analog of the subalgebra of a generated by sq1 and sq2 homotopy theory. While our main interest is the case when the group is. A survey of equivariant stable homotopy theory gunnar carlsson receiced 15 march 1991. We extend to this equivariant setting the main foundational results of motivic homotopy theory. This theory is quite newly formed, and there is a dearth of concrete results in it.
Equivariant homotopy theory has been a fundamental component of algebraic topology since its inception. Motivic analogues of mo mso university of michigan. Equivariant motivic homotopy theory 3 asoks program aso, 2. Salch the ravenel conjectures in nonclassical settingsequivariant, motivic, andor. It is a result which is the direct analogue of the pontryaginthom construction in classical and equivariant homotopy theory.
Equivariant and motivic homotopy theory isaac newton. In the process of my research, i realized it would be possible to construct motivic analogues of unoriented and oriented cobordism, which i refer to as mglo and mslo. Galois equivariance and stable motivic homotopy theory american. The site throughout this paper, we shall work over a base field k of characteristic 0. In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. Equivariant homotopy theory is homotopy theory for the case that a group g acts on all the topological spaces or other objects involved, hence the homotopy theory of topological gspaces. Equivariant and motivic stable homotopy theory, equivariant betti realization. This book introduces a new context for global homotopy theory. The tom dieck splitting theorem in equivariant motivic. I the homotopy limit problem for karoubis hermitian ktheory 25 was posed by thomason in 1983 48.
This leads to a theory of motivic spheres s p,q with two indices. Brown representability in equivariant motivic homotopy theory. Conjugacy class homotopy class nilpotent element equivariant cohomology obstruction theory these keywords were added by machine and not by the authors. We give a method for computing the c 2equivariant homotopy groups of the betti realization of a pcomplete cellular motivic spectrum over. According to topological stable equivariant homotopy theory, we distinguish between naive and genuine equivariant motivic spectra, inverting the integer indexed mo. Over the past week at least, urs and i and some others have been trying to understand equivariant stable homotopy theory from a highercategorical point of view, with some help from experts like peter may, john greenlees, and megan shulman. We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks x g, where g is a linearly reductive linear algebraic group. Dec 01, 20 in this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories.
The second author was supported by the ihes, the mpi bonn and a grant from the nsa. This conference will focus on the interplay and parallelisms between equivariant and motivic homotopy theory, providing a forum in which researchers can share insights and techniques from both disciplines. The goal of the workshop is to bring young researcher from both communities closer together. The results on motivic real cobordism are in ection 6. In particular, we show that, up to reindexing, the towers agree for all spectra obtained.
Peter may, equivariant homotopy and cohomology theory, cbms regional conference series in mathematics, vol. Prominent examples of this are equivariant stable homotopy, equivariant ktheory or equivariant bordism. The field has become more active recently because of its connection to algebraic ktheory see also. Y in the algebraic homotopy theory, to wit the a1homotopy theory of mv99. Michael andrews mit aravind asok university of southern california bert guillou university of kentucky. More generally, we show that betti realization presents the c 2 equivariant pcomplete stable homotopy category as a localization of the pcomplete cellular real motivic stable homotopy category. While our main interest is the case when the group is pro nite, we discuss our results in a more general setting so.
The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the sullivan conjecture that emphasizes its relationship with classical smith theory. In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. There is a canonical map from algebraic hermitian ktheory to the z 2homotopy fixed points of algebraic ktheory. While our main interest is the case when the group is profinite, we discuss our results in a. Equivariant motivic homotopy theory for finite flat group scheme actions was first defined by voevodsky in 9 in order to study motivic eilenbergmaclane spaces. Motivic homotopy theory of group scheme actions illinois. In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. An introduction to equivariant homotopy theory groups consider compact lie groups g and their closed subgroups h. The basic framework of equivariant motivic homotopy theory.
Here g is a nite group that acts on a small category i. Motivic analogues of mo and mso university of michigan. Workshop equivariant, chromatic, and motivic homotopy theory. During the period septemberdecember 2002 a research programme entitled new contexts for stable homotopy theory was staged at the isaac newton institute for mathematical research. Isaksen uniqueness of motivic homotopy theory following schwede. In chapters 23 of my thesis, i construct mglo and mlso and give a concrete description of the homotopy groups of each of them.
Unfortunately, the online part of the discussion has been taking place simultaneously in two different, hardtofind, and arguably. The new model structure introduced here will be compared to those by voevodsky and hukrizormsby. In the middle part of this thesis, the corresponding stable equivariant motivic homotopy theory is developed to some extent, allowing a later application. The relative picard group and suslins rigidity theorem 47 lecture 8.
At 23 sep 2015 galois equivariance and stable motivic homotopy theory j. T1 galois equivariance and stable motivic homotopy theory. Responsible for this website university of oslo library. C 2equivariant stable homotopy from real motivic stable homotopy mark behrens and jay shah abstract.
Equivariant homotopy theory has been a fundamental component of. Equivariant cycles and cancellation for motivic cohomology. Focus this conference will focus on the interplay and parallelisms between equivariant and motivic homotopy theory, providing a forum in which researchers can share insights and techniques from both disciplines. In equivariant stable homotopy theory for a nite group g, unpublished work of strickland str10, see chapter 3, contains a partial classi cation of thick ideals in the category shg f. The main ingredient of our method is developing gequivariant motivic stable homotopy theory for a.
The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology see there for details called bredon cohomology. Equivariant stable homotopy theory over some topological group g g is the stable homotopy theory of gspectra. The six operations in equivariant motivic homotopy theory. N2 for a finite galois extension of fields lk with galois group g, we study a functor from the gequivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence. Isaksen compute motivic stable homotopy groups at odd primes.
In the context of motivic homotopy theory, this was first studied by heller and ormsby in ho16, where the authors established a connection between c 2. Andrew blumberg, equivariant homotopy theory, 2017 pdf, github. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect a 1 homotopy theory is at least as complicated as classical homotopy theory. We compare the motivic slice filtration of a motivic spectrum over spec k with the c 2 equivariant slice filtration of its equivariant betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. We are now in the modern era of stable homotopy theory, with current topics such as topological modular forms and its variants, motivic stable homotopy theory, the study of commutative ring spectra and their localisations, galois extensions of ring spectra, and equivariant versions of most of those topics. We begin with the undations of equivariant unstable motivic homotopy theory. This process is experimental and the keywords may be updated as the learning algorithm improves. Workshop on motivic and equivariant homotopy theory. Equivariant homotopy and cohomology theory cbms regional. In this work, we present an alternative equivariant motivic homotopy theory, based on a slight variation of a nisnevichstyle grothendieck topology on the category of smooth gschemes over a eld. Equivariant stable homotopy theory has a long tradition, starting from geo. Motivic homotopy theory was introduced by voevodsky voe98.
Minicourse on motivic homotopy theory, by matthias wendt as the title suggests, the goal of the lectures is to provide a tour through some of the basic constructions in motivic homotopy, slightly geared towards the recent applications in algebraic classi cation problems. The field has become more active recently because of its connection to algebraic k theory. December 26, 2014 communicated by alexander merkurjev abstract. The second part of the thesis, which consists of one paper, is about the equivariant homotopy theory of socalled gdiagrams. The six operations in equivariant motivic homotopy theory arxiv. Equivariant stable homotopy theory 5 isotropy groups and universal spaces.
Equivariant motivic homotopy theory gunnar carlsson and roy joshua abstract. In the context of motivic homotopy theory, this was first studied by heller and ormsby in ho16, where the authors established a connection between c 2 equivariant homotopy theory and motivic. In 15, hu, kriz, and ormsby following notes of deligne 6 introduced equivariant stable motivic homotopy theory, and motivic real ktheory an analogue of atiyahs kr to solve thomasons homotopy limit problem on algebraic hermitian ktheory. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant. Lectures on equivariant stable homotopy theory contents. Mackey functors, km,ns, and roggraded cohomology 25 6. Thick ideals in equivariant and motivic stable homotopy.
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