The thread of ideas started with the work of poincar. A good i my view from the modern language the best reference is simplicial homotopy theory by goerss and jardine. How similar is this category to the ordinary category. Notes for a secondyear graduate course in advanced topology at mit, designed to introduce the student to some of the important concepts of homotopy theory. K1local homotopy theory, iwasawa theory and algebraic ktheory. Our starting point is the following result of hovey and sadofsky see 9. In addition to these notes, the basic source material for the course is the book local homotopy theory, by j.
Here global refers to simultaneous and compatible actions of compact lie groups. It arises in number theory as the progroup ring of a certain galois group, and in homotopy. Finally, i provided a brief summary of the extremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme interactions between homotopy theory and algebra. It has been noticed since the beginnings of equivariant homotopy theory that certain theories naturally exist not just for a particular group, but in a uniform way for all groups in a speci c class.
A speci c choice of such a continuous choice of local orientations is called a global orientation of mn. In algebraic geometry and algebraic topology, branches of mathematics, a 1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The first class consists of certain mapping cones of maps between wedge products of moore spaces. An application of algebraic rlocal homotopy theory hans scheerer mathematisches institut, freie universitiit berlin.
This result states that, in certain cases, homotopy groups of suspensions of a space stabilize. Let kn be a morava ktheory, let mbe a spectrum which is knlocal, and let gbe a. Sections 7 and 8 describe in detail two basic examples of model categories, namely the category top of topological spaces and the category ch rof nonnegative chain. These spectra are related by a canonical norm map nm. This book introduces a new context for global homotopy theory. Aravind asok usc connectedness in the homotopy theory of algebraic varieties. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely alge braic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory and algebraic ktheory.
Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, nonabelian cohomology, stacks, and local stable homotopy theory. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. Homotop y equi valence is a weak er relation than topological equi valence, i. The links below are to pdf files for my lecture notes for a course on local homotopy theory, or the homotopy theory of simplicial sheaves presheaves and presheaves of spectra. A survey of computations of homotopy groups of spheres.
Xequivalences, and it is the sense in which motivic homotopy theory replaces the unit interval of topology with the a ne line. Homotopy theory, topological spaces, simplicial sets, di erential. This lecture should introduce the notion of an oriented pdivisible group and the associated tempered cohomology theory, following section 4 of. Local homotopy theory there is a model structure on simplicial presheaves respectively, and quillen equivalently, simplicial sheaves on the site schj k et, for which the weak equivalences are those maps x. To make this precise, we use the theory of covering spaces. Lubintate theory, character theory, and power operations 1. On the homotopy theory of k local spectra at an odd prime by a. Internal and local homotopy theory zhen lin low 1 may 2014 abstract there is a wellestablished homotopy theory of simplicial objects in a grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of l. There is one of these categories for each natural number nand it.
A further advantage of this approach, which employs the familiar twosided cobar construction, is that it generalizes seamlessly to the enriched context. Let kn be a morava ktheory, let mbe a spectrum which is knlocal, and. There is a natural map from homotopy to singular homology, called the. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a. Let kn be a morava k theory, let mbe a spectrum which is kn local, and. As such, there are applications in topology, geometry, and number theory. Our thesis is that the galois group of a stable homotopy theory is a natural invariant that one can attach to it. In sections 710 model category theory is much more prevalent. Assuming basic knowledge of algebraic geometry and homotopy theory, local homotopy theory will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences. Introduction to higher homotopy groups and obstruction theory. Categorical homotopy theory mathematics johns hopkins. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important.
The links below are to pdf files for my lecture notes for a course on local homotopy theory, or the homotopy theory of simplicial sheaves presheaves. We will almost certainly fail to do justice to any of these ideas in the time available. The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed. This book consists of notes for a second year graduate course in advanced topology given by professor whitehead at m. Chief among these are the homotopy groups of spaces, specifically those of spheres. A classical 1groupoid is a structured collection ofmorphisms. Topological resolutions in k2 local homotopy theory at the prime 2 irina bobkova and paul g. Preceding the four main chapters there is a preliminary chapter 0 introducing some of the basic geometric concepts and. For our purposes the \homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. Lectures on local homotopy theory the links below are to pdf files for my lecture notes for a course on local homotopy theory. In this subsection, we work in the category of locally compact pointed. Discrete models for the plocal homotopy theory of compact lie groups and pcompact groups to be the pcompleted nerve l. The theory is due to fabien morel and vladimir voevodsky.
Elementary methods of calculation excision for homotopy groups. On the homotopy theory of klocal spectra at an odd prime. This is a brief introduction to how the lubintate moduli space, from algebraic geometry, appears in chromatic stable homotopy theory, via the goersshopkinsmiller theorem. Local homotopy theory mathematical association of america. References and resources for learning chromatic homotopy. Extending recent results on rlocal homotopy theory, we demonstrate that mild rreduced hopf algebras up to homotopy over can be modeled by mild rreduced differential graded lie. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopy theoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9. Browns representability theorems show that homology and cohomology are also contained in classical homotopy theory. Prominent examples of this are equivariant stable homotopy, equivariant ktheory or equivariant bordism. An application of algebraic rlocal homotopy theory.
Global equivariant homotopy theory studies such uniform phenomena, i. Investigates some surprising duality phenomena in the world of knlocal homotopy theory. This book introduces a context for global homotopy theory. This leads to a theory of motivic spheres s p,q with two indices. The choice of underlying topology determines the local nature of the resulting homotopy theory.
On the homotopy theory of klocal spectra at an odd prime by a. This is somewhat paradoxical considering that defining homotopy groups is very straightforward. Local homotopy theory department of mathematics faculty of. Connectedness in the homotopy theory of algebraic varieties. Homotopy theory is an important subfield of algebraic topology. This monograph on the homotopy theory of topologized diagrams of spaces. Local homotopy theory is the study of diagrams of spaces or spectrumlike objects and weak equivalences, where the weak equivalences are determined by a topology on the underlying index category. The category of topological spaces and continuous maps3 2. Homotopy theory is a subdomain of topology where, instead of considering the category. Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence. In homotopy theory, spaces are glued together fromdiscs. The resolution is built from spectra of the form ehf 2where e is the. Here kn denotes the nth morava k theory, one of the most important objects in the ordinary stable homotopy category.
A1homotopy theory from an 1,1categorical viewpoint thomas brazelton june 2018 abstract these notes are adapted from the homotopy theory summer school, berlin 2018, and are based on lectures given by florian strunkand georg tamme, as well as recitation sections by. This is an introduction to type theory, synthetic topology, and homotopy type theory from a categorytheoretic and topological point of view, written as a chapter for the book new spaces for mathematics and physics ed. Local homotopy theory department of mathematics faculty. Lecture notes on local homotopy theory local homotopy theory.
When studying the relationship to topological spaces now it depends on how much homotopy theory you already know. An application of algebraic r local homotopy theory, journal of pure applied algebra 91 1994 329332. The author has given the reader a fine introduction to homotopy theory in this book, and one that still could be read even now, in spite of the developments in homotopy theory that have taken place since the book was published 1975. Mostly finished, though it is a bit rough in places. There is a wellestablished homotopy theory of simplicial objects in a grothendieck topos, and folklore says that the weak. Stable homotopy theory is homotopy theory in the case that the operations of looping and delooping are equivalences as homotopy theory is the study of homotopy types, so stable homotopy theory is the study of stable homotopy types. Various ways to provide a home for global stable homotopy types have previously.
In order to approach the homotopy theory of a kplocal e1 local spectrum x, we first consider the homology groups kpx or. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. Homological codimension of modular rings of invariants and the koszul complex smith, larry, journal of mathematics of kyoto university, 1998. The next step is to bous eld localize the local model structure with respect to a rational point. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. This note contains comments to chapter 0 in allan hatchers book 5. Hsk, but this is not the motivic homotopy category we are after. This is the homotopy theory of simplicial sheaves, simplicial presheaves and presheaves of spectra. A survey of computations of homotopy groups of spheres and. The second one is a plocal version of the class of coformal spaces from rational homotopy theory. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by. Together with the lichtenbaumquillen conjecture, these were the original motivating principles and problems which led to the introduction and development of local homotopy theory in its present form.
Local homotopy theory is a foundational tool for motivic homotopy theory, and for the theory of topological modular forms in classical stable homotopy theory. Discrete models for the plocal homotopy theory of compact lie groups and p compact groups to be the pcompleted nerve l. Goerss abstract we provide a topological duality resolution for the spectrum ehs 1 2 2, which itself can be used to build the k2local sphere. Notation and some standard spaces and constructions1 1. Simplicial homotopy theory, and more generally the homotopy theories associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological algebra, algebraic geometry, number theory. As voevodskys work became integrated with the community of other researchers working on homotopy type theory, univalent foundations was sometimes used interchangeably with homotopy type theory, and other times to refer only to its use as a foundational system excluding, for example, the study of modelcategorical semantics or. This theory has found very remarkable applications. Over a subring r of q, we show that a certain differential graded lie ralgebra model is a complete r local homotopy invariant, for cw complexes satisfying some dimension and connectivity hypotheses. Introduction to higher homotopy groups and obstruction theory michael hutchings february 17, 2011 abstract these are some notes to accompany the beginning of a secondsemester algebraic topology course. In chapter 6, we give a short introduction to chromatic homotopy theory. An application of algebraic r local homotopy theory hans scheerer mathematisches institut, freie universitiit berlin. Preceding the four main chapters there is a preliminary chapter 0 introducing.
Global wellposedness for a system of kdvtype equations with coupled quadratic nonlinearities bona, jerry l. Goerss abstract we provide a topological duality resolution for the spectrum ehs 1 2 2, which itself can be used to build the k2 local sphere. The 2sphere s2 can be built from a point and a 2disc, but. Homotopy groups definitions and basic constructions.
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. Introduction to combinatorial homotopy theory institut fourier. Motivic homotopy theory ias school of mathematics institute for. Therefore, in homotopy type theory, when applying the substitution property, it is necessary to state which path is being used. This leads to the concept stable homotopy groups, and a whole stable homotopy theory. This entry is a detailed introduction to stable homotopy theory, hence to the stable homotopy category and to its key computational tool, the adams spectral sequence. In topology, two continuous functions from one topological space to another are called homotopic from greek. At the end, we will briefly extend some of these results to a higher chromatic setting. Local homotopy theory university of western ontario. In homotopy type theory, however, there may be multiple different paths, and transporting an object along two different paths will yield two different results. Although one can state the basic properties of homotopy colimits and limits without using model. In this paper we investigate the algebraic structure of the stable homotopy category localized with respect to khomology theory at an odd prime, and we give a purely algebraic classification of all homo topy types in that localized category. Lecture notes on local homotopy theory local homotopy.
Fix n0 and consider the category of kn local spectra, much studied by hopkins and coauthors, for example in hms94. Topological resolutions in k2local homotopy theory at the prime 2 irina bobkova and paul g. Chapter 1 the homotopy theory of simplicial sets 1. Y which induce weak equivalences of simplicial sets in all stalks i call these local weak equivalences, and for.
735 748 161 453 536 812 181 446 1428 668 112 800 616 572 733 1007 1029 285 995 144 858 956 136 231 474 215 368 1035 647 599 1229 1488 538 120 1462 1252 101