algebraic topology from a homotopical viewpoint - springer

Algebraic Topolog from a Homotopical Viewpoint Marcelo Aguilar Samuel Gitler Carlos Prieto... Springer New York Editorial Board North America: S... Carlos Prieto Algebraic Topology from

Algebraic Topolog from a Homotopical Viewpoint

Marcelo Aguilar Samuel Gitler Carlos Prieto

Springer

New York

Editorial Board (North America):

S Axler F.W Gehring K.A Ribet

Editors (North America): S Axler, F.W Gehring, and K.A Ribet

(continued after index)

Carlos Prieto

Algebraic Topology from a Homotopical Viewpoint

Auton a de Mexico Rochester, NY 14627-0001 Autonoma de Mexico

04510 Me DF US 04510 Mexico, DF Mexico gitler@math.rochester.edu Mexico marcelo@math.unam.mx cprieto@math.unam.mx Editorial Board

(North America):

S Axler F.W Gehring

Mathematics Department mathematics Department San Franco State

sity University of Michi,

San Francisco, CA 94132 Ann Arbor, MI 48109 1109 USA USA

K.A Ribet

Mathemat tics Departmen

University of California Berkeley

Berkeley, CA 94720-3840

USA

Mathematics Subject Classification (2000): 55-01

Library of Congress Cataloging-in-Publication Data

Aguilar, M A (Marcelo A.)

IH Title

QA612 A37 2002

§14’2—dc21 2002019556 ISBN 0-387-95450-3 Printed on acid-free paper

© 2002 Springer Werag New ‘York, Inc

All rights reserved Tt t lated pied in whol in part with

coftware ar hy <imil Lceienil Lod \ forhidd

the, are subject to proprietary rights

Printed in the United States of America

21 SPIN 10867195

E

www.springer-ny.com

Mottin OCmb Springer-Verlag New York Berlin Heidelberg

To Viola

To Sebastian and Adrian

hie hack ; 1 hed £ lool 1 1 y-theoretical methods x mm We bel 1 I 1 = h all 1 t 7 algebra Afi he f | h 5 using homotopy ‘oup: on

we

‘e define homology eroups Furthermore, with the same tools, > Blenders

1 These, in turn, ar e the

1 Lichad 1 1 1 xelated topics K+] To finish the book, 1 he PR fy tl 1 f 1 col 1 1 account of spectra

} 1 of the book tl f of the Dold-Thom

1 cy pie 1 Tarail

tions, is presented

1 lan} 1 led £ 1 and algebra I he book 1 1 1 £ 1 ad und aleal topological concepts are needed

74 13 lí bh whi +

1 £ 1 tha N TT] EM

vi PREFACE

b of Stenhen Sontz, to whom 1 thank Our

¢ 1 1 £ thọn zalnaabll

" my helned toi he Fnolicl 1 f the book Tts title

1 Toahn Mi] A bĩel 1 1 ] Lio} Taded 1 Last, but not least, ish t Ì ledge tl pport Albrecht Dold | fi he S 1 Jt

Autumn 2001 Samuel Gitler

Carlos Prieto!

PREFACE

INTRODUCTION BASIC CONCEPTS AND NOTATION

1 FUNCTION SPACES

1.3 The Exponential Law

2 CONNECTEDNESS AND

2.2 Homotopy Classe: 2.3 Topological Group:

2.5 The Fundamental Group 2.6 The fundamental Group of the Circle

1 là S 138

5 CW-COMPLEXES AND HOMOLOGY 149 5.1 CW-Complexes 2 020.002.020.020 002 2000.% 149 Infinite S ic Prod 167 5.3 Homology Group: 176

6 HOMOTOPY PROPERTIES OF CW-Co 189 6.1 Mac I 1M S 189 6.2 Homotopy Excision and Related Resuls 193

RELATED TOPICS 227 7.1 Cohomology Groups .0 0.202020 0048

A PRooF oF THE DoLD-THOM THEOREM 421 A1 Cc ¢ cp ats 421 A.2 Symmetric Products .02.- 431 A.3 Proof of the Dold-Thom Theorem 434

B PROOF OF THE

BotTT PERIODICITY THEOREM 437 B.1 A Convenient Description of BUxZ A437 B.2 Proof of the Bott Periodicity Theorem 440 REFERENCES 457

The fund lid £ alzel 1 1

A(f) - RCX) AEP A? h(Ÿ h len prop Gf 4] } 1 1 th are ho- í isomorphic to A(Y) 1 - A I I 1 £ 1der 1 Ít h (botl 1

= S” or if re is an H- group When Xp = S” we obtain the homotopy groups Tn(Y) =

Y] 1 1 ba£ 1 1 we

calculate It is | A 1k) 4} Togiaal abeli ra 1

mple homotopical structure So we associate to Y the free topological

1 ra Thy i 1h the 1 f Y acting a:

the zero element) T id is tl he infi

i t TP (Y spy) Ty 4.1 we are led to associate

| t P ] + The | T r1 FE 1 > 1

1 1 on Shyrati 1 a 1 cohomology, as well as K-theory

Tzabnai Cally far the dat F+baT 1 theory (see [42, 43]) On the oth 1, Voevodsky [79] and ott 1

1 toi non £ fie hol ] talsel tonol to alwel + 74 he found fo 141

VOCVOQdSKR

£ he Mil 1 Lin } Mĩ] K+} Cad Band the Cal 1 £

TM, P Real X > qe] t 1 1 1 Le] Lie} £ lool Te deaf 144 wed od 1 homology theories Ti ao 1 tứ he Eilent 1 Mac Te | al 1 I for KT Tacboad 1 A] of thịc boel 1 coe cay 1

hy lie of aleat 1 T Prank Ad 1 1 he Hoant + probl Tưng tỊ ly Tưng Tri] borat 1 Tr isely §, S', SẼ, and Sĩ or,

le 1 1 dives lool 1 ls the comple numbers, the quaternions, and the Cayley numbers Throughout the text

tha 1 £ bund] Lich a fy]l cl « devoted

ha} 1 £ 1t tr dof

4 tont 1 led “noi 1 fl 1

is required Altl h fi ly ] ; no

1 led 1 1 1£ 1 4} lof 1 1 1 + 1; len to obtam the £ sa] : duced invariants

he d t fo]] XÃ 1 1

1 1 1] 11 1 yy h of Lich je divided ; 1 1 1isHrnaniehed bự 1o]

) Exerci ince many of them are

1 14_T1 Thom

1 sey ae yee 171 1

B 1 periodici 1 I hen freal 1i 1 of di In the appropriate 1 let

1 1 44 1 ea} ] tư +] applications

1 1] f P The former is, in a way, the basic

Ÿ trhe hich celai all id “aye in 1 bool 1 built We study the degree of 1 mail

and suspensions are carefully studied

1 tay + dự of có] T CHR P 1 Tueed in CH E 12T 1 Tema

1 oy ye 1 iewed Tl Vad in C1 1 he Lick Further homotopy “of the Đai

Mt 1 ws 1 Thiet wy J) in 4}

£Ị tthe MỸ 1 the Filen} Mac T

£ 1 Tnceif 1

1 1 1e fned sazl 1 1 Jott KA

1 :bla t† 1 1 ented, but not yet proved Later on Ck he Hont it 10, tk Ad 4] ti 1

Lo leu mm á 1

on the spheres S°-!

1 Jatianchin } line band] a) 1

en the fact that the classify; ] 1 bundles, namely RP and CP to +I f+he T1 1 Eilenk £ Mac L 1 hand] A simple

TỊ 1 1 1] 1efne + +t] Wh; 1 1 bund] 1 the Cl f P 1 vector bundles We finish tl t of lized f the col bool h Ck 11 1 1

he B 1 a 1 he 4 £ spectra end the chapter

1 1 A eA 1 1 tot +] los B 1 mÁ 1

numbered (X.1.1 1.2 where X is either A or B)

A 1 te alphabetical ind jor chanld foal f look ¢ 1 bói

font 1

is R or C

ca] 1 14 1 1 1

in the text If X i logical 1A X, in agreement wi e

] chin y 1+1 3A

x u Yd ] pological f X and Y On the other hand, if

product, that is, Hel or "ls J (2, 2) Likewise, if A c Vi is a subspa ace,

e At —

complement of A tin ụ ith ] SOME BASIG “OPOLOGIOAL SPAOES all play an important role for us

point) {0} C

Euclidean n-space, such that

= (@1, ,%n) | a ER, 1<é<n} Using the equality

`" am ốc ify the ©: 1 R R” with R™” Likewise, we identify

R 1 h the " closed sul Liat R 1 R +, We give Urmo Rr = R@® the hort]

ing the equalit

I z denotes TontiE tl L e iP fz =z +1 Úp to the to chow that tl} and IK two norms coincide

TG hall he fallout 1 £Toelid space:

D” = {x € R”| |x| < 1}, the unit disk of dimension n S*! = {x € R” | |x| = 1}, the unit sphere of dimension n — 1 D” = = {x €R”| |x| < 1}, the unit cell of dimension n

T=T!'= |0,1]C R, the unit interval Briefly, call D” the unit n-disk, S' —! the nit (n — 1)-s, sphe the umát n- neal Vand rm the unit n- oube ti is worth etining that " of

e connected), except for S 1 i - ic, of 1 The disks, the spheres, 1 fl + for the 0-cell D =* `

antipodal action, that 1 is, C be =—-#€ S The orbit space of the ae action,

fact, the inclusion S*-! ¢ S” induces an inclusion RP*-' c RP” and the

n °°, RP” coincides topologically with RP*

On the other hand, the circle group St: = {¢ € C| |l¢]| = 1} acts on Sẽn+1 C Ertl} 1 ye

(C21) ,0%n41) T f titying z€ sien with €z€ sả, for all ge si, is s denoted by Cr” and i is q

dimen 2n ) T The action of sie on S +1 _ induce an action on Ss * who e orbit space i denoted by ce II ject¿

In anal ] , the inelusion 8?°~1 C §?“*!, defined by the inclusion C? c C71, induces an inclusion CP°-! c CP” and the union

Um CP” coincides topologically with CP nants are not zero The subgroup O GL, {R) (U, GL,,(C)) consisting

and U,;=S!

SOME GENERAL BASIC CONCEPTS

then ker(f) = {g € G |

AA its image An arrow of the form ‘—=* represents an inclusion °

group monomorPhisth, and finally, one of the form ——> represents an

Atp 3c

is called exact at B if im(f) = ker(g)

1 Leht] ] £ laced embed dy

€ A ® Ái CDA, b> is where hi, = hy" © _ In

For each 2 we have homomorphism: Boe Ta hi: A; lim Tai gi : tì Lee iti 1

>—>Ð 4; —> colim A;

We have, as in the topological case, that

hk oki, = hi: A; —> colim A; The aloahral Tit alen I he follow; is:

ag, ) = (a, — h? (ap), ay — h3(ag), a3 — h3(aa),.-.) :

We define its émzé as the kernel of d, lim A‘ = ker(d) , and its derived limtt as the cokernel of d,

lim! A‘ = coker(d) = (H 4) /im(d)

In this way we obtain an exact sequence lim A’ —+ [| 4’ —: |4 —: lim'4' — 0

ohitio -ohk_.), that hbo fp = fy: :B— A forallk>i>1, then there exists a unique homomorphicta f: B— lim A’ such that hyo f = fi

this is expressed

i 1 of limit, and one denotes it by the

CATEGORIES, FUNCTORS, AND NATURAL ‘TRANSFORMATIONS

Associativity If f: A—> B,g: B —+ C,andh: C —> D, then ho(gof)=(hogjof: ASC

if f: , , g: , then golp=g

lowing:

1 +; 1 f ts Set(A, B) is the set of function from A to B

27 ¢ 1 £ 1£

£ 1 1 1 Led 1 1 Jor]

3 7 ¢ 1 £ 1£

£ (abel 11 1 lef bai Thy a

4 TI £ 1 £ 1

1 1 £ 11 1

1 1 £ 1> lned functions

6 Ad ˆ

£ A 1 1 The 4

= 1 Lott 1 £ hie det £ 1 ¢ Let T,,T: :C —3 D be

¢ tứ her] bon -

A I fi from T; to Ts, in symbols y : T; —+ Ts, assigns

t bat biect £ A of bị T Ì fA B TCA Tr(A) of D 1€ *) OF t nl following diagrams is commutative

ng ne) nye)

“nA cp Rte, “iA go Bh), according to whether T;, T:

We present two result

n this : : : ¢

a «smooth bump function Namely, given A C V C R” where A is closed and V is open in R”, a bump function of 4 in V is a continuous function h:R' —> =0

ø(1

ay = 4 — t) a(1—t) + a(t)’ which is such that

Ba =1 ift <0, 0<Ø@)<1 #0<¿<1, Ba) =0 ift>1

d let J D k ball; that i r Then for x € R” the function

bt} ¢,

Let now U ° R” be open and bounded, and let Vv = R” be such that V5 TI —>Ro

V Let Di} h that 7D Dic V and let A; be a smooth bump function of D; in Di Define h by

h(x) = 1—(1— hif2)) + + (1— hel)

1 hy : : : * Smooth approximation theorem Let U CR” be open, and let f: U—

Let moreover

|g(e) — F{®)| for abix g J Ww

TT of 1 le the Vi 1 lar] ed 1 bại

p(#) — ƒ(œ)| < e for all ø U and take a smooth bump function h of W' in W" Then define

g(z) = h(s)p(œ) + (1— h(z))f(œ) for z € D, Then g is smooth in W UW’, o|W! = 2|W’, gl U — W“ = ƒ|D — W*, and

|ø(#) — ƒf(z)| < € for allze wi

ooth deformation theorem Let U Cc R” and V C R” be bounded open sets and let p : U —> a continuous map Take W,W! c R™ such that W CW! CW' CU Then there exists a map w:U that:

ớ (1) yw — > V is smooth Q) |U — W1 = ý|D — W* and ý © pre (U-W% The proof is as follows C 1 W') by afi

W, 0|U — W1 = u|U — W", and bá) ‘pice forall ev Then the linear deformation

me ĐT = " — ee) - ee)

TT yr Le., it is s relative 1 to U- W“ In particular, vio) cy

are regular

The following result holds " [57]

where U C R™ Ef, in particular m<n, then s0)" pen, then = ;

Brown-Sard theorem Let y : U —> R” be a smooth map, where U C R” is open Th 1 f regul , to 2e d „ TP sults, one has the following theorem

m 2 LetU Cc and VỀ C R” be bounded open sets and let yp:

—> V be a continuous map Take W, W en such that W

We CU Then there exists a map > : U —> V such that

(QQ) |U — W* = ylU-—W' andy ~ prel (U — (3) There is a point y € V such that p(y) ts a p manh (m—n)-mantfold, and in particular, ifm <n, then vy) = PARTITIONS OF UNITY

alues Œ, or in some vector space For example, it 1s an ezerc¿se to

pr hat {fy :

function f:

ry

FUNCTION SPACES

Tunet P The #

ill be made in this text Tl i f this ck

i + k f tl E f f i P 1 We shall assume a +] fond in +l texts [27, 34, 60, 83], for example 1.1 ADMISSIBLE TOPOLOGIES

: £ : 1 Togiasl 1 have useful properties

1.1.1 DEFINITION Let X,Y be sets We denote by Y~ the set of functions

foll hall 1 ] in M(X,Y) We consider the evaluation map

el: such that e'(f,z) = f(x), and its restriction

e: M(X,Y)x X > Y 1.1.3 DEFINITION W Tus : that a topol in M(X,Y) is admissible if tl

Ị BP fA ] pol f Y and that generalizes the product topology

1.2.1 DEFINITION Tl topol in M(X_YV)} hy the family of sets

continuous and elf, k) = - /Ú) < Us there exist — VY, of f in

The family {W,} f f K, which is compact, so that

1 co: Leamily Ui Ww b that KT UW, Let

1 U,i=l, Put V=Vin - nV Then fe and V0 0% ince itgeV a nd ke K, then k = W for some i So, = e(g,k) € e(V x W2) C eV; And this shows that U* is open in MAX, ”,

ith compact closure V such that x € } y W

no Œ,z)<U uffices to that UY xV ø ev) Indeed, if f/ € UY and 2! € V, then 0) eU, thet

1.3 THE EXPONENTIAL LAW

4 1 FuNcTION SPACES

To realize this, it suffices to define

pi BY (BY by el A(a)(y) = flea) and, as its inverse,

Pi (BY 3 2 by va) (ay) = s(œ)0)

@: M(X xY,Z) — M(X,M(Y, 2)

is continuous

rst statement, ] 1 f 1 Y-3Xx¥Y5zZ, where i.(y) = (x,y), which clearly is continuous (Note that if X = @, the proposition is me

ertion, let L suffices to show that a tf) (0) is s open ín x So take rE adr wo

Ũ } finite © subfamily Vi,.- uch that f(W Ve that covers K U Then Wi Put W = Win i -OWm, where lofzin X

We claim that WoC Xu (UX) Indeed, if z' S W a e kK, ‘nen

el fe) (k) = ơi k), but k € V; for some i, and 2’ € Đa So 5N, ke

b: M(X,M(Y, Z)) — M(X x Y,Z)

is well defined Let g : X —+ M(Y,Z) be continuous It suffices to show

ith compact closure V such that

CV CW Therefore, g(x)(V) CU, and so g(x) € UY, which is open

in M Họ: Z }

@: M(X x ¥, 2) 3 M(X, M(Y,2))

is a homeomorphism

M(X, My Z)) if U is open Í 1K and X and Y mepetinl (cf [27 TLB] or 1.3.4 below) Then K x L is compact, and

M(X x Y,Z), t L FUE Xx L ;

M(Y,Z) —› M(X,2) such that f#(g) = go f Similarly, if g : Y —> Z is continuous, then it

gự: M(X,Y) —> M(X, 2) such that gu(f) = 1X ƒ# and gy are, in fact, continuous

usually denote M(I, I; x Zo) by Q(X, 29) or, if the I is obvi from context, by QX (cf 1.3.9 further on)

We define their product to be the pair

1 TÔ ÔÔÓÔ : b that (TP CO here Lo Ví ZL ;Ơi 944,20} 1i uch that tol } = £o 1.3.9 DEFINITION The space M(I”,0I”; 3 lled tk

CONNECTEDNES SA ALGEBRAIC INVARIANTS

We shall study the which

2.1 PATH CONNECTEDNESS

of a path in a topological space X 2.1.1 lati DEFINITION ft Let Ũ I 1 A TY We define the following ha and a( ) = =ự W 1 darth 1} below) X

each pair of points 2,y € X

2E1 also, O-connected, if x ~ y for