Various ways to provide a home for global stable homotopy types have previously. Rational homotopy theory of automorphisms of manifolds. Disconnected rational homotopy theory andrey lazarev and martin markl abstract. Rational homotopy theory is the study of spaces up to rational homotopy equivalence. Arkowitz book is a valuable text and promises to figure prominently in the education of many young topologists. Gis a pair of adjoint functors, and suppose that f and g. In contrast to previously encountered situations, model theoretic techniques are intrinsically insu. In generality, homotopy theory is the study of mathematical contexts in which functions or rather homomorphisms are equipped with a concept of homotopy between them, hence with a concept of equivalent deformations of morphisms, and then iteratively with homotopies of homotopies between those, and so forth. Stable homotopy theory in problems in homotopy theory. In homotopy theory, spaces are glued together fromdiscs. Rational homotopy theory and deformation problems from.
On the topological side this means that rational global homotopy types need not decompose. A classical 1groupoid is a structured collection ofmorphisms. The name of the subject comes from the use of concepts from abstract algebra, such as groups, to study topological spaces. Rational homotopy theory serre, 1950s quillen, 1969 sullivan, 19731977 padic homotopy theory adams, mass. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. X, but two such loops are regarded as determining the same element of the fundamental group if one loop can be continuously deformed to the other within the space x. Final functors in unenriched category theory 101 8. Request pdf the universal fibration with fibre x in rational homotopy theory let x be a simply connected space with finitedimensional rational homotopy groups. The theory of rational homotopy is the study of spaces with rational equivalences. Well stick with the rst one, which will be our main example throughout this course. Felix wierstra koszul duality and rational homotopy theory. Introduction homotopy theory is the study of topological spaces with homotopy equivalences. This is meant to expand on the notion of fundamental group and is a powerful tool that can be used for distinguishing spaces. Rational homotopy theory yves felix, stephen halperin, j.
Rational homotopy theory is the homotopy theory of rational topological spaces, hence of rational homotopy types. The idea is that instead of looking at the cochains which can be stricti ed to a cdga, we look at the homotopy groups of a rational. Rational homotopy theory alexander berglund november 12, 2012 abstract these are lecture notes for a course on rational homotopy theory given at the university of copenhagen in the fall of 2012. For example, use oliviers thesis strasbourg 20 to understand the shimomura schools calculations at large primes. These homotopy groups are very closely related to corresponding rational. Instead, a rather intricate blend of model theory and classical homotopy theory is required.
Unlike the usual encoding in terms of the set theory this one respects equivalences. All formulations give notions which are equivalent in a homotopy theoretic sense, see slide 110 below, although the objects involved look very different in detail. Its aim was to highlight areas in which category theoretic abstractions provide a particularly valuable insight into classical homotopy theoretic constructions. Wu in homotopy theory, one can decompose hspaces as a product of homotopy indecomposable hspaces, and cohspaces as a wedge of homotopy indecomposable cohspaces. There are even looser notions of equivalence, for example, two spaces are qequivalent if their rational homologies are equivalent. Probabilityone homotopy maps for tracking constrained. These notes are from a series of lectures given at the workshop on the homotopy theory of homotopy theories which took place in caesarea. Distinguishing homotopy types older work rational homotopy theory. The stable rational cohomology of arithmetic groups was computed by borel in 15. We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on di. Abstract in this article, we study the elementary and basic notions of homotopy. For further information on rationalization, the reader is refered to section 9 of 18. As an application, we show for a large class of simplyconnected nonnegatively curved. Rational homotopy theory in arithmetic geometry, applications to rational points christopher david lazda june 5, 2014 supervised by dr ambrus p al submitted in part ful lment of the requirements for the degree of doctor of philosophy in mathematics of imperial college london and the diploma of.
Sullivans work on the subject was preceded by quillen, who showed that the homotopy theory of rational spaces can be described in terms of the homotopy theory of di erential graded lie algebras over q. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. An elementary illustrated introduction to simplicial sets. Abstract sullivans seminal paper, in nitesimal computations in topology, includes the application of his techniques to nonsimply connected spaces, and these ideas have been used frequently by other authors. A brief introduction to homotopy theory hedayatzadeh. Unstable homotopy theory in problems in homotopy theory. Categorical tools for homotopy colimit computations 97 8. In this short note, we construct a class of models of an extension of homotopy type theory, which we call homotopy type theory with an interval type. Notation and some standard spaces and constructions1 1. Rational homotopy types of simply connected spaces can be identified with. Disconnected rational homotopy theory sciencedirect. Recall that a homeomorphism is given by two maps f.
Lisez le livre rational homotopy theory and differential forms directement dans votre navigateur. In part i we exhibit a chain of several categories connected by pairs of. This simplification of homotopy theory makes calculations much easier. 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. Rational homotopy theory is the study of rational ho motopy types of spaces and of the properties of spaces and maps that are invariant under rational homotopy equivalence. Obstructions to nonnegative curvature and rational. 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. In mathematical logic and computer science, homotopy type theory hott h. The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere sn, for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of sn are trivial and that the third homotopy group of s2 is also. An introduction to stable homotopy theory semester project by maximilien holmbergperoux responsible professor prof. This book introduces a context for global homotopy theory. Rational homotopy theory lecture 7 3 where p is an acyclic bration, and where the dotted arrow exists since f is a co bration since the compositions u.
Lectures on homotopy theory, volume 171 1st edition. Other readers will always be interested in your opinion of the books youve read. Homotopy theory considers even more geometric objects as equivalent objects. Other parts of the subject are homology theory and cohomology theory. Rational homotopy, complex manifolds, complex projective varieties, hodge theory, weight spectral sequence, formality. We construct a natural transformation from the bous eldkuhn functor e.
The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. With its modern approach and timely revisions, this second edition of rational homotopy theory and differential forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory. Jan 18, 2014 cannon and conner developed the theory of big fundamental groups. Global equivariant homotopy theory studies such uniform phenomena, i. We describe a category, the objects of which may be viewed as models for homotopy theories. Theoretically there is a collection of atomic hspaces or atomic cohspaces. A pair c,e is a relative category, and from the point of view of tpair homotopy theory is relative category theory. This book consists of notes for a second year graduate course in advanced topology given by professor whitehead at m.
The universal fibration with fibre x in rational homotopy. Rational splitting of global k theory as mentioned above, not every rational global functor or outopmodule is projective. 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. This note contains comments to chapter 0 in allan hatchers book 5. For stable homotopy, the suspensions of all spaces are cohspaces. Q, the lusternikschnirelmann category catg, the cohomology eulerpoincar e characteristic. Rational homotopy theory and deformation problems from algebraic geometry john j. So the derived category does not split as in the case of rational mackey functors. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of.
We will use basic results from rational homotopy theory for which 7 has become a standard reference. I am trying to read the paper rational homotopy theory by quillen and am stuck with the notion of complete augmented algebra. The category of topological spaces and continuous maps3 2. Rational homotopy theory and differential forms phillip. Before sullivan, quillen developed another algebraic model of rational homotopy theory in terms of dg lie algebras. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a bit for the. Change of base for homotopy limits and colimits 99 8.
Nov 30, 2016 homotopy theory, which we have barely scratched the surface of in this post, is just one part of the subject called algebraic topology. The idea of the fundamental group cornell university. Rational global homotopy theory and geometric xed points. Pdf gratuit rational homotopy theory and differential. Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way.
Generalized homology theory and stable homotopy theory 240 17. Y is a map between rational spaces, then f is a rational homotopy equivalence if and only if f is a weak equivalence. Rational homotopy theory and differential graded category. There are two seminal papers in the subject, quillens 20 and sullivans 25. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. The thesis mainly focuses on the sullivan equivalence, which models rational spaces by commutative differential graded algebras contrary to quillens. Mandell iu cochains and homotopy theory aug 31 6 24.
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. Updated content throughout the book, reflecting advances in the area of homotopy theory. Jun 07, 2014 in my first year at harvard, i had an opportunity to teach a graduatelevel topics course entitled categorical homotopy theory. Homotopy theory of higher categories by carlos simpson. M ill son department of mathematics, university of maryland, college park, md 20742, usa this paper is a description of research i have been doing over the last four years, applying some of the methods and ideas of rational homotopy theory as. The bousfieldkuhn functor and topological andrequillen cohomology mark behrens and charles rezk abstract. Using the tools of homotopy theory, we can examine what kind of theory we get if we allow ourselves such a notion of equivalence. Modern machine learning problems typically have multiple criteria, but there is currently no systematic mathematical theory to guide the design of formulations and exploration of alternatives. Note that a weak equivalence is always a rational equivalence. The thread of ideas started with the work of poincar. Via the classical adjoint functors between the categories of commutative differential graded. In its initial phase research in rational homotopy theory focused on the identi of these models. Rational homotopy theory ii yves f elix, steve halperin and jeanclaude thomas world scientic book, 412 pages, to appear in march 2012.
Master thesis on rational homotopy theory, the study of homotopy without torsion. Introduction to homotopy theory is presented in nine chapters, taking the reader from basic homotopy to obstruction theory with a lot of marvelous material in between. I would like to thank the organizers of the summer school on \ rational homotopy theory and its interactions for inviting me to give this course in rabat. This result of quillen was the impetus for later work of. For instance, in homotopy theory, a solid ball of any dimension and a point are considered as equivalent, also a solid torus and a circle are equivalent from the point of view of homotopy theory. A classi cation theorem theorem jordan, 1866 two orientable surfaces with boundaries are homeomorphic if and only if they have the same genus and the same number of boundary curves. Rational homotopy theory lecture 22 benjamin antieau 1. This book introduces a new context for global homotopy theory. For our purposes the \ homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. 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. Friedlander examines basic topology, emphasizing homotopy theory. An elementary illustrated introduction to simplicial sets greg friedman texas christian university. The 2sphere s2 can be built from a point and a 2disc, but.
It was founded by dennis sullivan and daniel quillen. We take a moment to explain why this might be useful for understanding the above motivating questions. That section contains the statement and proof of many of the major theorems of classical homotopy. Complex cobordism and stable homotopy theory 242 21. The idea of homotopy can be turned into a formal category of category theory. Introduction to higher homotopy groups and obstruction theory. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. Homotop y equi valence is a weak er relation than topological equi valence, i. It is based on a recently discovered connection between homotopy theory and type theory. Rational homotopy theory and differential forms springer. The focus of the conference, and subsequent papers, was on applications of innovative methods from homotopy theory in category theory, algebraic geometry, and related areas, emphasizing the work of younger researchers in these fields. Homotopy theory of higher categories from segal categories to ncategories and beyond. G, the koszulpoincare series u gz and the formal dimension fdg.
In this part we give an application of the developed theory to the structure of mc spaces and describe the second version of disconnected rational homotopy theory based on dglas. Rational homotopy theory and differential forms springerlink. Quillens approach to rational homotopy theory see 17, 6, bous eld and kans work on completions, localization, and limits in homotopy theory 1, quillens abstract treatment of homotopy theory 18, and various aspects of. Rational homotopy theory is the study of the rational homotopy category, that is the category obtained from the category of 1connected pointed spaces by localizing with respect to the family of those maps which are isomorphisms modulo the class in the sense of serre of torsion abelian groups. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. 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. They have many applications in homotopy theory and are necessary for the proofs in section 6. Included is a discussion of postnikov towers and rational homotopy theory. Sullivan model and rationalization of a nonsimply connected space homotopy lie algebra of a space and fundamental group of the rationalization, model of a fibration holonomy operation in a fibration malcev completion of a group and examples lusternikschnirelmann category depth of a sullivan lie algebra growth of rational homotopy groups structure.