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http://www.springer.com/series/223

Introduction to Homotopy

Martin Arkowitz

Theory

�

Department of Mathematics

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Printed on acid- free paper

Springer is part of Springer Science+Business Media (www.springer.com)

ISBN 978- 1- 4419- 7328- 3

Library of Congress Control Number: 2011933473

Mathematics Subject Classification (2010): 5

Springer New York Dordrecht Heidelberg London

Martin Arkowitz

Dartmouth CollegeHanover, NH 03755-3551USAMartin.A.Arkowitz@Dartmouth.edu

e-ISBN 978-1-4419-7329-0DOI 10.1007/978-1-4419-7329-0

5Q05, 55R05, 55S35, 55S45, 55U30

e-ISSN 2191-6675ISSN 0172-5939

© Springer Science+Business Media, LLC 2011

To Eleanor, my bashert,

and to

Dylan, Jake, and Gregory Arkowitz

Preface

This book deals with homotopy theory, which is one of the main branches ofalgebraic topology. The ideas and methods of homotopy theory have pervadedmany parts of topology as well as many parts of mathematics. A general ap-proach in these areas has been to reduce a geometric, analytic, or topologicalproblem to a homotopy problem, and to then attempt to solve the homotopyproblem, usually by algebraic methods. Thus, in addition to being interest-ing and important in its own right, homotopy theory has been successfullyapplied to geometry, analysis, and other parts of topology. There are severaltreatments of homotopy theory in general categories. However we confineourselves to a study of classical homotopy theory, that is, homotopy theoryof topological spaces and continuous functions. There are a number of booksdevoted to classical homotopy theory as well as extensive expositions of it inbooks on algebraic topology. This book differs from those in that the unify-ing theme by which the subject is developed is the Eckmann–Hilton dualitytheory.

The Eckmann–Hilton theory has been around for about fifty years butthere appears to be no book-length exposition of it, apart from the earlylecture notes of Hilton [40]. There are advantages, both expository and ped-agogical, to presenting homotopy theory in this way. Dual concepts occur inpairs, such as H-space and co-H-space, fibration and cofibration, loop spaceand suspension, and so on, and so do many theorems. We often give completedetails in describing one of these and only sketch its dual. This is done whenthe latter can essentially be derived by dualization. In this way we shortenthe exposition by reducing the amount of repetitious material. This also al-lows the reader to test his or her understanding of the subject by supplyingthe missing details.

There is another advantage to studying Eckmann–Hilton duality theory.Frequently the dual of a result is known or trivial. But from time to timethe dual result is neither of these and is in fact an interesting problem. Thiscould give the reader material to work on.

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viii Preface

A feature of this book is that it is designed primarily for students to learnthe subject. The proofs in the text contain a great deal of detail. We alsotry to supplement the discussion of several of the concepts by explainingthem intuitively. We provide many pictures and include a large number ofexercises of varying degrees of difficulty at the end of each chapter. Theexercises that have been used in the text are marked with a dagger p:q andthe more difficult exercises are marked with an asterisk p�q. It is generallyregarded as important to do the exercises in order to learn the material. Ithas been said many times that mathematics is not a spectator sport.

This book has been written so that it can be used as a text for a universitycourse in algebraic topology. We assume that the reader has been exposed tothe basic ideas of the fundamental group, homology theory, and cohomologytheory, material that is often covered in a first algebraic topology course.We state explicitly the results from these areas that we use and summarizethe essential facts in the appendices. The text could also be used by math-ematicians who wish to learn some homotopy theory. However, the book isnot intended to introduce readers to current research in topology. There aremany texts and survey articles that do this. Instead it is hoped that this bookwill provide a solid foundation for those who wish to work in topology or tolearn more advanced homotopy theory.

We now summarize the text chapter by chapter. The first chapter containsa discussion of the notion of homotopy and its variations and related notions.We consider homotopy relative to a subset, homotopy of pairs, retracts, sec-tions, homotopy equivalence, contractibility, and so on. Most of these shouldbe familiar to the reader, but we present them for the sake of completeness.If X and Y are based spaces, we define the homotopy set rX,Y s to be theset of homotopy classes of based maps X Ñ Y. Next CW complexes are in-troduced and some of their elementary properties established. These spacesplay a major role in the rest of the book. Finally, there is a short sectionindicating some of the reasons for studying homotopy theory.

The next chapter deals with grouplike spaces and cogroups. The formeris a group object in the category of based spaces and homotopy classes ofmaps. The latter is the categorical dual of a group object in this category.We consider loop spaces and suspensions, important examples of grouplikespaces and cogroups, respectively. This leads to a discussion of basic prop-erties of the homotopy groups rSn, Y s, where Sn is the n-sphere. We thendefine and construct spaces with a single nonvanishing homology group, calledMoore spaces, and spaces with a single nonvanishing homotopy group, calledEilenberg–Mac Lane spaces. These give rise to homotopy groups with coeffi-cients and to cohomology groups with coefficients. This gives a homotopicalinterpretation of cohomology groups. The chapter ends with a discussion ofEckmann–Hilton duality.

In Chapter 3 we discuss two dual classes of maps, fiber maps and cofibermaps. Fiber maps are defined by the covering homotopy property which is awell-known feature of covering spaces and fiber bundles. Cofiber maps appear

Preface ix

often in topology because the inclusion map of a subcomplex of a CW complexinto the complex is a cofiber map. A fiber map E Ñ B determines a threeterm fiber sequence F Ñ E Ñ B, where F E is the fiber over the base pointof B. A cofiber map i : A Ñ X determines a cofiber sequence A Ñ X Ñ Q,where Q � X{ipAq is the cofiber. We then study fiber bundles. We giveexamples of fiber bundles and these provide many examples of fiber sequences.We conclude the chapter by showing that any map can be factored as thecomposition of a homotopy equivalence and a fiber map or as the compositionof a cofiber map and a homotopy equivalence.

The next chapter deals with exact sequences of homotopy sets. The mainsequences are a long exact sequence associated to a fiber sequence and oneassociated to a cofiber sequence. By specializing these sequences we obtain theexact homotopy sequence of a fibration and the exact cohomology sequenceof a cofibration. We next study the action of a grouplike space on a spaceand the coaction of a cogroup on a space. These give additional informationon the exact sequences of homotopy sets. We then consider homotopy groupsand define the relative homotopy groups of a pair of spaces. We discuss theexact homotopy sequence of a pair and the relative Hurewicz homomorphism.We conclude the chapter by introducing certain excision maps which are usedin Chapter 6.

Chapter 5 is devoted to some applications of the exact sequences of thepreceding chapter. We begin with two universal coefficient theorems. The firstrelates the cohomology groups with coefficients of a space to the integral co-homology groups of the space and the second relates the homotopy groups ofa space with coefficients to the homotopy groups. Then we show how the op-eration of homotopy sets in Chapter 4 can be specialized to yield an operationof the fundamental group π1pY q on the homotopy set rX,Y s. This operationis used to compare the based homotopy set rX,Y s with the unbased homo-topy classes of maps X Ñ Y. Finally we calculate some homotopy groups ofseveral spaces including spheres, Moore spaces, and topological groups.

Chapter 6 contains the statement and proof of many of the importanttheorems of classical homotopy theory such as (1) the Serre theorem on theexact cohomology sequence of a fibration, (2) the Blakers–Massey theoremon the exact homotopy sequence of a cofibration, (3) the Hurewicz theoremswhich relates homology and homotopy groups, and (4) Whitehead’s theoremregarding the induced homology homomorphism and the induced homotopyhomomorphism. In the first part of the chapter we define homotopy pushoutsand homotopy pullbacks and derive some of their properties. A major resultthat is used to prove both the Serre and Blakers–Massey theorems is that acertain homotopy-commutative square is a homotopy pushout square.

In Chapter 7 we discuss two basic and dual techniques for approximatinga space by a sequence of simpler spaces. The obstruction theory developed inChapter 9 is based on these approximations. The first technique, called thehomotopy decomposition, assigns a sequence of spaces Xpnq to a space X suchthat the ith homotopy group of Xpnq is zero for i ¡ n and is isomorphic to

x Preface

the ith homotopy group of X for i ¤ n. From the point of view of homotopygroups, the spaces Xpnq approach X as n increases. The second technique,called the homology decomposition, is similar with homology groups in placeof homotopy groups. We consider several properties and applications of thesedecompositions. In the last section of the chapter we generalize these decom-positions from spaces to maps.

In Chapter 8 we derive some general results for the homotopy set rX,Y s.We give hypotheses in terms of cohomology and homotopy groups that implythat the set is countably infinite or finite. We consider some properties of thegroup rX,Y s when X is a cogroup or Y is a grouplike space. We show thatif Y is a grouplike space, then rX,Y s is a nilpotent group whose nilpotencyclass is bounded above by the Lusternik–Schnirelmann category of X.

In the final chapter we consider two basic problems for mappings. In thefirst, called the extension problem, we seek to extend a map defined on asubspace to the whole space. In the second, called the lifting problem, weseek to lift a map into the base of a fibration to a map into the total space.These are two special cases of the extension-lifting problem. We develop anobstruction theory for this problem which gives a step-by-step procedure forobtaining the desired map. We present two approaches to the theory. Forthe first, we take a homotopy decomposition of the fiber map and assumethat the desired map exists at the nth step. This determines an element in acohomology group, whose vanishing is a necessary and sufficient condition forthe map to exist at the pn�1qst step. In the final section we discuss a methodfor obtaining obstruction elements by taking homology decompositions. Theseelements are in homotopy groups with coefficients.

After Chapter 9 there are six appendices. These are of two types. Onetype consists of results whose proofs in the text would be a digression of thetopics being treated. The proofs of these results appear in the appendix. Theother type provides a summary and reference for those basic results aboutpoint-set topology, the fundamental group, homology theory, and categorytheory that are used in the text. Definitions are given, the results are stated,and in some cases the proof is either given or sketched.

In conclusion, I would like to acknowledge the many helpful suggestionsof the following people: Robert Brown, Vladimir Chernov, Dae-Woong Lee,Gregory Lupton, John Oprea, Nicholas Scoville, Jeffrey Strom, and DanaWilliams. I would like to express my appreciation to the following people atSpringer: Katie Leach for editorial assistance, Rajiv Monsurate for advice onTex, and Brian Treadway for drawing the figures. Finally, I am particularlyindebted to Peter Hilton for having introduced me to this material and tu-tored me in it while I was a graduate student. To all these people, manythanks.

Contents

1 Basic Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Spaces, Maps, Products, and Wedges . . . . . . . . . . . . . . . . . . . . . 21.3 Homotopy I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Homotopy II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 CW Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6 Why Study Homotopy Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . 29Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 H-Spaces and Co-H-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 H-Spaces and Co-H-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Loop Spaces and Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4 Homotopy Groups I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.5 Moore Spaces and Eilenberg–Mac Lane Spaces . . . . . . . . . . . . . 602.6 Eckmann–Hilton Duality I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Cofibrations and Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2 Cofibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4 Examples of Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.5 Replacing a Map by a Cofiber or Fiber Map . . . . . . . . . . . . . . . 104Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 The Coexact and Exact Sequence of a Map . . . . . . . . . . . . . . . . 1164.3 Actions and Coactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.4 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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xii Contents

4.5 Homotopy Groups II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Applications of Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 Universal Coefficient Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 Homotopical Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . 1605.4 Applications to Fiber and Cofiber Sequences . . . . . . . . . . . . . . . 1635.5 The Operation of the Fundamental Group . . . . . . . . . . . . . . . . . 1695.6 Calculation of Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . 177Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6 Homotopy Pushouts and Pullbacks . . . . . . . . . . . . . . . . . . . . . . . 1956.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.2 Homotopy Pushouts and Pullbacks I . . . . . . . . . . . . . . . . . . . . . . 1966.3 Homotopy Pushouts and Pullbacks II . . . . . . . . . . . . . . . . . . . . . 2076.4 Theorems of Serre, Hurewicz, and Blakers–Massey . . . . . . . . . 2146.5 Eckmann–Hilton Duality II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7 Homotopy and Homology Decompositions . . . . . . . . . . . . . . . . 2337.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.2 Homotopy Decompositions of Spaces . . . . . . . . . . . . . . . . . . . . . . 2347.3 Homology Decompositions of Spaces . . . . . . . . . . . . . . . . . . . . . . 2477.4 Homotopy and Homology Decompositions of Maps . . . . . . . . . . 254Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

8 Homotopy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678.2 The Set rX,Y s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678.3 Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2708.4 Loop and Group Structure in rX,Y s . . . . . . . . . . . . . . . . . . . . . . 275Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9 Obstruction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2839.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2839.2 Obstructions Using Homotopy Decompositions . . . . . . . . . . . . . 2849.3 Lifts and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2889.4 Obstruction Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

A Point-Set Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

B The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

C Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Contents xiii

D Homotopy Groups of the n-Sphere . . . . . . . . . . . . . . . . . . . . . . . . 312

E Homotopy Pushouts and Pullbacks . . . . . . . . . . . . . . . . . . . . . . . 314

F Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Hints to Some of the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Introduction to Homotopy TheoryPrefaceContents