Smith previously, given an hcobordism w with a speci ed handlebody presentation, we had an asso. In the last lecture, we asserted that xand y are simply homotopy equivalent if and only if. An immediate result of this theory is that, if two spaces have the same simple homotopy type, they have the same homotopy type. Algebraic ktheory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.
A homotopy equivalence is called simple if it is homotopic to a homotopy equivalence induced by a sequence of elementary expansions and collapses. This note contains comments to chapter 0 in allan hatchers book 5. Simple homotopy equivalence of nearby lagrangians abouzaid, mohammed and kragh, thomas, acta mathematica, 2018. Notation and some standard spaces and constructions1 1. Hatcher while much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. 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. The basic structure we shall use for such a theory, a dspace, is a topological space equipped with a family of. Siebenmanns infinite simple homotopy theory s and hatchets higher simple homotopy theory h are developed along these geometric lines. The authors intention is to rely on the geometric approach by appealing to the readers own intuition to help understanding. Printed in great britain homotopy, simple homotopy and compacta steve ferryi received 5 may 1978. The first to successfully give an algebraic model for rational homotopy theory was quillen in the 1960s. Discrete morse theory from a simplehomotopy point of view. The category of topological spaces and continuous maps3 2.
Equivariant stable homotopy theory 5 isotropy groups and universal spaces. A second reason for writing the book is pedagogical. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple homotopy theory. 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. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory rethinking the homotopy hypothesis old the analytic homotopy theory of 1groupoids is equivalent to that of spaces. Einfacher homotopietyp homotopieaquivalenz homotopy. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps.
We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. 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. For example, if we imagine a stretchy object, then all the shapes we can stretch or twist it into are homotopy equivalent. Kamps, tim porter, abstract homotopy and simple homotopy theory, world scientific 1997. 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. This monograph introduces the category of spaces that are boundedly controlled over the usually noncompact metric space z. However, formatting rules can vary widely between applications and fields of interest or study. So, with this motivation, whitehead introduced the concept of simple homotopy equivalence between simplicial complexes, that later was generalized for cw complexes, spaces created by himself. Boundedly controlled topology foundations of algebraic. Two cwcomplexes are simplehomotopy 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. It presents elements of both homology theory and homotopy theory, and includes various applications. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a.
Similarly to classical simple homotopy theory, the strong. The category of topological spaces and continuous maps. So you are starting in historically the first place anyway. They form the rst four chapters of a book on simplicial homotopy theory. A space is contractible if it is homotopy equivalent to a onepoint space. A course in simplehomotopy theory marshall m cohen springer. Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed. Abstract homotopy theory abstract homotopy and simple. To simplify cwcomplexes it uses the same elementary homotopy equivalences that generate simple homotopy equivalences. 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. Controlled simple homotopy theory and applications ebook.
This book introduces a new context for global homotopy theory. 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. Simple homotopy theory and whitehead torsion richard wong abstract. A course in simplehomotopy theory graduate texts in. The most prevalent idea for extending simple homotopy theory. A course in simplehomotopy theory, graduate texts in mathematics, no. Higher simple homotopy theory lecture 7 september 17, 2014 recall that nite polyhedra xand y are concordant if there is a piecewiselinear bration q. Controlled simple homotopy theory and applications springerlink. This is a lecture on a historical motivation for looking at algebraic k theory. We say that there is an elementary collapse from l to k.
Homotopy equivalence is an equivalence relation on spaces. Various ways to provide a home for global stable homotopy types have previously. These notes were used by the second author in a course on simplicial homotopy theory given at the crm in february 2008 in preparation for the advanced courses on simplicial methods in higher categories that followed. A simplehomotopy approach to the finiteness obstruction. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simplehomotopy theory.
An immediate result of this theory is that, if two spaces have the same simplehomotopy type, they have the same homotopy type. We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. So, with this motivation, whitehead introduced the concept of simplehomotopy equivalence between simplicial complexes, that later was generalized for cw complexes, spaces created by himself. Controlled simple homotopy theory and applications book. Global equivariant homotopy theory studies such uniform phenomena, i. That section contains the statement and proof of many of the major theorems of classical homotopy. New the synthetic theory of 1groupoids ismodeled byspaces. A homotopy is a deformation change of form of one thing into another without cutting it. Introduction to combinatorial homotopy theory institut fourier. Pdf abstract homotopy and simple homotopy theory semantic. A homotopy type is an equivalence class of homotopy equivalent spaces.
Discrete morse theory is used to understand how cwcomplexes can be simpli ed without changing the homotopy type. Statement of results the homotopy pexponent of a topological space x, denoted exppx, is the largest e such that some homotopy group ix contains a zpesummand. Homotopy theory is a subdomain of topology where, instead of considering the category of. System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. What is the simplest category which could help our homotopy theorist understand at least some of the notions appearing in such a chapter. Pdf elements of homotopy theory download full pdf book. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. Algebraic k theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. An example of a homotopy this is a supplement to chapter 24 of scienti. Homotopy simple english wikipedia, the free encyclopedia. Controlled simple homotopy theory and applications. This is an excellent subject for a topology student to grow up on. The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories e. Differential topology from the point of view of simple homotopy theory.
This is a lecture on a historical motivation for looking at algebraic ktheory. Whiteheads theory of simple homotopy types is inspired by tietzes theorem in combinatorial group theory, which states that any. Introduction to higher homotopy groups and obstruction theory. To simplify cwcomplexes it uses the same elementary homotopy equivalences that generate simplehomotopy equivalences. Homotopy types and simple homotopy types lecture september 30, 2014 in the rst part of this course, we introduced a simplicial set m whose nsimplices are nite polyhedra e n r1for which the projection map e. One famous example is the homotopy equivalence of a coffee cup and a donut. In the last lecture, we asserted that xand y are simply homotopy equivalent if and only if they are concordant, and proved the \if direction. Simple homotopy types and finite spaces internet archive. Geometric, algebraic, and arithmetic objects are assigned objects called kgroups. Here is the homotopy type of the most simple space. Chapman t simple homotopy theory for anrs 167 horneomorphism, they. 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.
It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. Imagine a simple minded homotopy theorist, or even better a homotopy theory android trying to understand the introductory, settheoretic, chapter commonly preceding undergraduate texts in topology, say. Whiteheads simple homotopy theory aimed to give a combinatorial description of ho motopy types of simplicial complexes. In the present article we describe constructions of model structures on general bicomplete categories. In mathematics, particularly the area of topology, a simplehomotopy equivalence is a refinement of the concept of homotopy equivalence. Mx of a differentiable manifold m with a cell decomposition. There is an elementary expansion from k to l if l k f d nwhere f. In mathematics, particularly the area of topology, a simple homotopy equivalence is a refinement of the concept of homotopy equivalence. A course in simplehomotopy theory graduate texts in mathematics this book grew out of courses which i taught at cornell university and the university of warwick during 1969 and 1970.
In the 30s50s, much of homotopy theory was focused on developing combinatorial ways to determine homotopy equivalences. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing. The important new work of waldhausen relating attheory is outlined, very briefly and imperfectly, in 7. 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. Model categories with simple homotopy categories jeanmarie droz and inna zakharevich abstract. The idea of homotopy can be turned into a formal category of category theory. The techniques developed in 1 are applied to the simple. Homotop y equi valence is a weak er relation than topological equi valence, i. 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. Homotopy, simple homotopy and compacta sciencedirect. Directed algebraic topology is beginning to emerge from various applications. This book consists of notes for a second year graduate course in advanced topology given by professor whitehead at m.
1059 1160 382 939 602 477 1515 1074 366 115 705 1118 98 118 399 1098 734 370 761 1151 1273 402 1395 1287 300 234 37 320 834 704 965 1249 1320 1198 108 149 16 609 166 1176 824 720 597 1235 1005 830 871