Ncohen a course in simple homotopy theory pdf

In mathematical logic and computer science, homotopy type theory hott h. Henri cohen written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Homotopy, homotopy equivalence, the categories of based and unbased space. This book grew out of courses which i taught at cornell university and the university of warwick during 1969 and 1970. Printed in great britain homotopy, simple homotopy and compacta steve ferryi received 5 may 1978. 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. This book grew out of courses which i taught at cornell university and the. A classical 1groupoid is a structured collection ofmorphisms. In mathematics, particularly the area of topology, a simplehomotopy equivalence is a refinement of the concept of homotopy equivalence.

This paper of course can only be a scratch on the sheer endless surface, how. The generality of the results implies of course that certain speci. A space is contractible if it is homotopy equivalent to a onepoint space. Homology can be described elegantly in this theory. A course in simplehomotopy theory by marshall m cohen, 9780387900568, available at book depository with free delivery worldwide. Persistence produces the same hierarchy but uses a simplified. Calculating the fundamental group of the circle in homotopy. The category theory and homotopy theory suggest new principles to add to type theory, and type theory can be used in novel ways to formalize these areas of mathematics. A mailing list for the spring 2014 hott book reading group at mit, which allows you to add yourself and links to archives. 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. Of course, the same reduction to chain com plexes is necessary in order to calculate whitehead groups and show that the theory is nonvacuous. Relative homotopy groups, homotopy fiber, long exact sequence in homotopy, whitehead theorem. Whiteheads theory of simple homotopy types is inspired by tietzes theorem in combinatorial group theory, which states that any. Acrobat reader a course in simple homotopy theory full text information a course in simple homotopy theory.

References in addition to the references listed at the end and in footnotes, i shall refer to the following of my course notes available at. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by cohen, moore, and the author, on the exponents of homotopy groups. Moduli spaces from a topological viewpoint semantic scholar. This is a lecture on a historical motivation for looking at algebraic k theory. That section contains the statement and proof of many of the major theorems of classical homotopy. Homotopy, simple homotopy and compacta sciencedirect. Introduction homotopy equivalence whitehead s combinatorial approach to homotopy theory. An undergraduate number theory course will also be helpful. A course in simple homotopy theory, graduate texts in mathematics, no. These notes were written for the summer school on \rational homotopy theory and its interactions july 2016, rabat, morocco. I wrote it because of a strong belief that there should be readily available a semihistorical and geo metrically motivated exposition of j.

Peter mays textbook a concise course in algebraic topology is probably the closest. A course in simplehomotopy theory marshall m cohen springer. In topology, a branch of mathematics, a collapse reduces a simplicial complex or more generally, a cw complex to a homotopy equivalent subcomplex. Finally, we will discuss lens spaces which provide examples of homotopy equivalent nite cw complexes which are not simple homotopy equivalent. Differential topology from the point of view of simple homotopy theory. A short course on the interactions of rational homotopy theory and complex algebraic geometry joana cirici abstract. This book consists of notes for a second year graduate course in advanced topology given by professor whitehead at m. It should not be taken with the expectation of fulfilling any course requirement or degree credit. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simplehomotopy theory.

The course offers an introduction to algebraic topology centered around the theory of higher homotopy groups of a topological space. This is an excellent subject for a topology student to grow up on. In, which is a textbook treatment of this and many other aspects of classical homotopy theory, fred gives no less than 5 equivalent formulations of this question. The category of topological spaces and continuous maps. Errata to rings, modules, and algebras in stable homotopy theory pdf surv 2. I could extend this reading course over a year and wouldnt mind studying hard so i would want to cover a fair bit of material. A homotopy type is an equivalence class of homotopy equivalent spaces. Finally, i provided a brief summary of the extremely fruitful interactions between rational homotopy theory and local algebra.

In the 30s50s, much of homotopy theory was focused on developing combinatorial ways to determine homotopy equivalences. Pdf a model structure for the homotopy theory of crossed. Recall from whiteheads simple homotopy theory the basic geometric. Introduction to combinatorial homotopy theory francis sergeraert ictp map summer school august 2008 1 introduction.

Introduction homotopy equivalence whitehead s combinatorial approach to homotopy theory cw complexes a geometric. However, a few things have happened since the book was written. Buy a course in simplehomotopy theory graduate texts in mathematics softcover reprint of the original 1st ed. Change of base for homotopy limits and colimits 99 8. A course in simplehomotopy theory marshall m cohen. A course in number theory and cryptography, neal koblitz. The theory of simplicial sets offers a model of homotopy theory without using topological spaces. A simplehomotopy approach to the finiteness obstruction. A wellprepared student will have some background in formal logic and some prior experience with simple type theory.

In algebraic topology, homotopy theory is the study of homotopy groups. I am very grateful to the students in that course for comments on earlier versions of these notes. Stacks and the homotopy theory of simplicial sheaves jardine, j. When good references are available we may not include the details of all the proofs. These notes are from a series of lectures given at the workshop on the homotopy theory of homotopy theories which took place in caesarea. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. In mathematics, particularly the area of topology, a simplehomotopy equivalence is a. All of the oznal pro perties of the theory functoriality, sum theorem, product theorem, etc.

To that end we introduce the modern tools, such as model categories and highly structured ring spectra. A course given by bob harper with videos and notes. In homotopy theory, spaces are glued together fromdiscs. 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. Categorical tools for homotopy colimit computations 97 8. Rational visibility of a lie group in the monoid of self homotopy equivalences of a homogeneous space kuribayashi, katsuhiko, homology, homotopy and applications, 2011. These groups offer more information than the homology or cohomology groups with which some students may be familiar, but are much harder to calculate. Two cwcomplexes are simple homotopy equivalent if they are related by a sequence of collapses and expansions inverses of collapses, and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map. Introduction to unstable homotopy theory 5 neisendorfer also introduced a homotopy bockstein spectral sequence to study the order of torsion elements in the classical homotopy groups. The theory of simple homotopy types has shown itself to be natural and basic for. This book contains a semihistorical and geometrically motivated exposition of j. A course in some aspects of classical homotopy theory. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a bit for the. Whiteheads beautiful theory of simplehomotopy types.

They have many applications in homotopy theory and are necessary for the proofs in section 6. Similarly to classical simple homotopy theory, the strong homotopy. At an intuitive level, a homotopy class is a connected component of a function space. A more abstract, but at the same time geometric, approach to simple homotopy theory was explored in cohens book as well as in the papers by eckmann, eckmann and maumary, and siebenmann, listed above. A model structure for the homotopy theory of crossed complexes. Most of the papers referred to are at least twenty years old but this reflects the time when the ideas were established and one imagines that the situation will be different in the second volume. A more abstract, but at the same time geometric, approach to simple homotopy theory was explored in cohen s book as well as in the papers by eckmann, eckmann and maumary, and siebenmann, listed above. Keywords eilenbergmac lane and moore spaces hspaces and cohspaces fiber and cofiber spaces homotopy homotopy and homology decompositions homotopy groups loops and suspensions obstruction theory pushouts and pull backs. Moduli spaces, mapping class groups, cobordism, algebraic ktheory of spaces. Global equivariant homotopy theory studies such uniform phenomena, i.

A course in computational algebraic number theory henri cohen a description of 148 algorithms fundamental to numbertheoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The idea of homotopy can be turned into a formal category of category theory. First we specify more closely the subcomplexes l, in which of course. This course can be viewed as a taster of the book on homotopy type theory 2 which was the output of a special year at the institute for advanced study in princeton.

The book could also be used by anyone with a little background in topology who wishes to learn some homotopy theory. Modern classical homotopy theory graduate studies in mathematics volume 127. The 2sphere s2 can be built from a point and a 2disc, but. These notes grew out of a graduate topology course i gave at stanford university during the spring term, 1998. I decided that the goal of this course should be to establish carefully the foundations of rational homotopy theory, then to treat more super. A survey of computations of homotopy groups of spheres.

Here is the homotopy type of the most simple space. Final functors in unenriched category theory 101 8. A course in simple homotopy theory by marshall m cohen, 9780387900568, available at book depository with free delivery worldwide. The subject is accessible as in the courses mentioned at the outset to students who have had a good one semester course in algebraic topology. A course in computational algebraic number theory henri. 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. Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed. Everyday low prices and free delivery on eligible orders. 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. Collapses find applications in computational homology. Various ways to provide a home for global stable homotopy types have previously.

Springer made a bunch of books available for free, these. Pdf elements of homotopy theory download full pdf book. Introduction to higher homotopy groups and obstruction theory. It would be nice if that course included some mention. This book introduces a context for global homotopy theory. 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. A course in simplehomotopy theory, graduate texts in mathematics, no. The objective of this course is to do several computations. Simple homotopy theory and whitehead torsion richard wong abstract. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory discs versus morphisms there is a fundamental mismatch. Topology is the branch of geometry that deals with largescale features of shapes.

The seminar should be of interest to those interested in algebraic topology. The homotopy theory associated to the mapping class groups is another. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Our principal goal is to establish the existence of the classical quillen homotopy structure, which will then be applied, in various ways, throughout the rest of the book. I have basic knowledge of algebraic topology corresponding to hatchers algebraic topology. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple homotopy theory. The aim of much of homotopy theory in the 1930s and 1940s was to try to lay. In this chapter we give a detailed introduction to the theory of lens spaces. This book introduces a new context for global homotopy theory. The hopf invariant problem we can often turn geometric problems into problems in stable homotopy theory, which gives us a rich set of tools to attack our problems. Collapses, like cw complexes themselves, were invented by j. The subject is accessible as in the courses mentioned at the outset to students who have had a good one semester course in algebraic.

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. Algebraic methods in unstable homotopy theory this is a comprehensive uptodate treatment of unstable homotopy. Homotopy equivalence is an equivalence relation on spaces. Of course, the morse function doesnt give rise to a unique. Homotop y equi valence is a weak er relation than topological equi valence, i.

One cliche is that a topologist cannot distinguish a doughnut from a coffee cup. Here global refers to simultaneous and compatible actions of compact lie groups. In chapter 6, we give a short introduction to chromatic homotopy theory. The writing bears the marks of authority of a mathematician who was actively involved in setting up the subject. A pair c,e is a relative category, and from the point of view of tpair homotopy theory is relative category theory. Whiteheads beautiful theory of simple homotopy types. Advanced topics in computational number theory henri. A course in simplehomotopy theory graduate texts in. In particular, they were interested in when a homotopy equivalence can be. Chapters one through five form a homogenous subject matter suitable for a sixmonth or yearlong course in computational number theory. Higher homotopy groups, weak homotopy equivalence, cw complex.