1\ ^ ON NUMERICAL HOMOTOPY INVARIANTS AND HOMOTOPY FUNCTORS by DIEN WEN CHEN B.Sc., N a t i o n a l T a i w a n U n i v e r s i t y , M.Sc., N a t i o n a l T s i n - H u a A THESIS THE SUBMITTED the 1966 F U L F I L M E N T OF FOR THE DEGREE OF DOCTOR OF in University, IN P A R T I A L REQUIREMENTS 1963 PHILOSOPHY Department of MATHEMATICS We accept required THE this thesis as conforming to standard UNIVERSITY OF B R I T I S H March, 1972 COLUMBIA the In presenting requirements of British it freely agree thesis of this for I scholarly that or copying gain partial that reference for or his may be of MATHEMATICS The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date the be of Library copying granted Columbia this of the University shall and S t u d y . allowed permi s s i o n . Department at representatives. publication not fulfilment the extensive purposes by shall degree agree for permission for in advanced available that financial an Columbia, my D e p a r t m e n t stood thesis I further of this by the It is thesis without my make Head under- for written Supervisor: Dr. Armin Frei Abstract The Functor" as We f i n d strong category, potency object BA-, which BX = B A X . such main and do not that the CatBX < C a t X , nilBX < which weak case where under where case consider B = A the are spaces X is the A of the F has for the based BA- to duals . we product invariants, Lusternik-Schnirelmann a co-group, smash the functor, n i l - i.e. WcatBX < WcatX union case of of A with we and the smash a point) actually that (in homotopy type of the this is We a l s o satisfy functor show then equivalent; in in spaces A category to have of CW-complexes) * X and / A a generalization investigate order the preserves XV(AX) of conditions the a properties we . Finally, that "Smash m Furthermore = ZBv(AZB) found the smash homotopy particular (disjoint + a co-H space, homotopy functor is BX = category. when X is paper X the numerical category, This based this a space catBX < c a t X , AxX/ L-S in nilX. We t h e n functor to various increase have study associates category, in of of we some collect of a few ii our counterexamples results are false. to show TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENTS iv Chapter 1 INTRODUCTION AND NOTATION 2 THE SMASH 3 THE FUNCTOR 4 COUNTEREXAMPLES 1 FUNCTOR A 4 ^ ~ AND SOME G E N E R A L I Z A T I O N S . . . 40 51 REFERENCES 62 i i i ACKNOWLEDGEMENTS The for his grateful author invaluable for Department of the the is deeply assistance financial indebted and of i v Professor encouragement. assistance University to of British the A. Frei He i s also Mathematics Columbia. 1 Chapter INTRODUCTION The "Smash Functor" product w cat in have X and Chapter Cat smash do strong that (in in BX < this to the weak case under X , cat nil X . paper is a space various increase BX < C a t nil study category, and not AND NOTATION associates We f i n d category, X the smash numerical homotopy category, Lusternik- when the X is a smash BX < c a t These the X , co-group) functor, w cat results BX < are given 2. In the as nil.potency, we of BA-, which such Schnirelmann i.e. object BX = BAX . invariants, the main 1 Chapter functor 3 we where consider B = A + a particular (disjoint example union of of A with Ax X a point) preserves category in which L-S of case BX = category. based /^ . F u r t h e r m o r e we spaces of the based C W - c o m p l e x e s ) when X i s a c o - H s p a c e , X V ( A X ) a r e homotopy e q u i v a l e n t ; t h i s A of Hilton's formula This xyR / A functor show that homotopy actually (in type the of AxX the space /^ and is a generalization a ZBV(AEB). In this chapter 2 we also in order show investigate to that have conditions the In Chapter the duals properties 4, of Throughout gory of based complexes. are by supposed * to F has to BA- satisfy . we found for collect a few counterexamples some of our paper having results we the otherwise preserve shall based stated, base work in maps which the type and is A and B be 1) use pointed 3) 4) A* following topological I = [0,1] with 2) the is the closed the free (where we do based space) path not consider PA i s the i.e. PA = It is a contractible ftA is the { U E based space then interval {ui e path space A u 1 | loop space A u(0) 1 | over A I a as over ( 0 ) = *} A, . space. over = A, o)(l ) = i.e. *} CW- homotopies always 0 = * is ftA = unit cateof notations. spaces, to false. homotopy all point, are . We s h a l l Let we this spaces Unless a functor denoted A V B is A and AB = B A X B product the one-point union . / A V B of = AAB A and is the means homeomorphic " means homotopic. " smash B. " = " * of and 4 Chapter Let J£ be basepoint-preserving space is, in it 3£ • Then any map f Bf : BX — - : BY also FUNCTOR the category continuous B- pointed maps, and a space BX t o space X Y 3* such that the is of in B l any , B- let with B be a fixed to 2£ . That X e 0^ , and associates a map and B(fo g) = Bfo preserves null maps and = 1 x functor it spaces B X have LEMMA i.e. SMASH from Clearly we THE a functor associates to 2 if then 2.1: a * 3 The (rel functor *) Ba = B 3 ( r e l Proof: Since a B- : X *) - preserves Y : BX — * 3 homotopy. (rel *) BY : . X — * Y , there exi sts H : X x I ^ Y with H(x,0) = a(x) H(x,l) = 8(x) Bg. 5 Consider the 1 B x X x diagram B x H I B x Y p x 1 BX x where we If it p,q have are b = * or above x H B b, maps. x For b, b e = * . Since p is an identification map and that p x lj there of is is Ba Lemma B3 2.1 . I is X, I — 3 - BY such IT i s the t e t compact, identification Obviously e map. that the required I we on spaces. an H" : BX x commutative. and operates topological a map also x bH x , bH(x , t) is B, H(x,t) then From actually identification , diagram homotopy BY x = * follows Therefore 1 q o I can see The Its that the homotopy objects are functor B- category of pointed spaces, based its I, 6 morphisms B- is are also based homotopy a functor homotopy type classes of on " U J * , of the homotopy CW-compl e x e s , X e J+ and let be the k+l copies consisting x. = * of for of X , and points (x least one at Clearly category of B e bJ* . cartesian product i ... of maps. provided k+l II X c~r Let continuous 1 let , x T k+l (X) C , 2 i(l x < i k + < k+l) nX 1 ) be the such subspace that . k+l Let A(x) = (x, A : X — » x , •• • • ,x) . II X be the diagonal map, i.e. Let k+l x be the the (k+l)-smashed / = power of T k + i ( x ) X with q : n X x (k+l) i d e n t i f i c a t i o n map. DEFINITION wcat (k+i) X < k) Iff 2.2: X has weak category < k (written q A « * . THEOREM 2 . 3 : Let wcat B , X be pointed BX < w c a t X . spaces, then 7 Proof: be proved; so, (a nonnegative If let us wcat X = 0° , assume t h a t then nothing wcat x < k By d e f i n i t i o n integer). k+l n x 2.2, needs with we to k finite have (k+l) 1 with q A - * , then Bq o BA = B(q o A) - B * = * Cons i d e r BX rk+1 n x BA N _Bg_ x (k+l) k (1) I k+l n x where h is the unique map d e f i n e d by the BX commutative BX (k+l) diagrams: 8 in which p i k : projections. + 1 n X =- X T h e map k i s and p > : Kn BX X — »v BX nV i defined as the composition _r e -it h a of maps : 6 1 X< +D k in which x t h e map 6 i s ( k + 1 ,_ ) E k+ and t B's and X ' s i n t h e smash the right i = 1 , 2, i n (1) proves diagram. q~oA we have (k+i) ^ J U product. is B of t | j :m ( k + 1 ) B X maps, ( k + i ) interchanges It is commutative. the order easily checked Furthermore, of that for ,k+l , p / o h o (BA) = which B t h e homeomorphism w h i c h square x the composition A^ n is (k+i) that ( B p . ) o (BA) = B ( p h o (BA) = A . o A) = l i Thus (1) is g a x = p.o A , commutative Since = q"o h o (BA) = k p B q o B A wcat BX < k . = k o B B ( q o A) = * , 9 DEFINITION topological property may If be no space that X is X has covered such 2.4: by the the a locally exists, finite for Cat out loss X = °° , of CW-complex complexes X H • X x I m Cat be then BX < C a t generality, we may assume is covered by k+l F u r t h e r m o r e we may , and — that X the * X of with a the a CW-complex which subcomplexes. connected and BX < C a t X . X is Cat certainly that X itself assume that B Withis self-contractible also let true X < k < °° . self-contracting respect m k > 0 Cat that which of Cat X = °° . assume m each Cat us X . to type X e k/* CW-complex, let integer then Let Since Proof: least category self-contractible THEOREM 2 . 5 : be strong homotopy k+l integer The a sub- * belongs homotopies . r Let H" : BX x I — BX be t h e h o m o t o p i e s m m m i n d u c e d by 1 x H : B x X x I — B x X , as i n t h e B m m m p r o o f o f Lemma 2 . 1 , t h e n H" : 1 ^ = * ( r e l *) . D m J m Clearly k+l BX = V J B X contractible m m cat , hence subcomplexes DEFINITION gory, X, of B 2.6: BX i s BX . The a topological covered by Consequently k+l self- Cat BX < k . Lusternik-Schnirelmann catespace X is the least integer I 10 k > 0 with the subsets which exists, then property are cat cat equivalent remark a that CW-complex, 2.6 by the by closed in be X. covered If present to G.W. previous classical then an we Whitehead, one when arguments, may subsets alternative DEFINITION if as there A = j in or replace even 2.7: Definition exists k+l such integer X is it the by a map that inl<J* . follows open n X , I Then <f> : X it is First we that subsets cat T k+l (X) C j is the + 1 i n c l u s i o n and X is Def. X < k if be and only k+l T (X) 1 such that diagonal map. (X) A in I n x i k if k+l n X o <j> , T L-S subcomplexes. Let 2.2. show k+l where open d e f i n i t i o n of and k+l fined no by X = °° . X , due to X may contractible We s h a l l category that is the de- 11 We d o n o t c l a i m alent lent throughout i. = { ( x , . . . . x . _ 1 let p : 1 we category , * , x 1 k+l n X *> Def.'s the category on t h e p o i n t e d and that 1 + 1 CT* , 2.6 and 2.7 a r e however they o f CW-complexes. ,----x k + 1 )|x j e are equivaIn f a c t let k+l c n x , ifj<k+i} x, equiv- . *- X be t h e projection, I I < £ < k+l . Then i f there exists c}> s u c h that A = j o <J> , have l x = pz o A « pz . Consider cjTMZ.) . Since T k + 1 (X) 1 form a covering with closed p i j (j) subsets CW-complex, o j o <j> k+l = \J z . , i =l these cj> (Z.) _1 1 of X. F u r t h e r m o r e we h a v e tp~ (Z.)j 1 which we h a v e Conversely, = * . Thus p^ j X i s covered are contractible i n X. cat X < k . k+l X = W X m=l m Then P^A = l by k+l Since cat X < k . suppose <P - X is a x , with X m homotopy which a subcomplex extension contracts X contractible property, m F < m < k+l . there X. Then, exists F : m by the X* I — m F (x 1 in Then m , 0) = ( x , l ) these x x e X - * F 's m x X e determine m m , a unique map k+l - such F F : X x n i 1 x that commutes. there { l» 2'"*-> m' F Now exists F(x,0) = A ( x ) <J>(x) == F ( x , l ) and such F(x,l) that A - e T j (X), o <J> . i . 13 THEOREM 2 . 8 : B , X e . (Based Proof: us assume exists <j> , Consider that such the When cat cat that X = °° , <j> = A sion is h is map, ~ defined and commutative. because Let as £ = h The X . the then conclusion by is trivial. definition there . diagram rk+1 n X I BX cat X < k < °° , j following BX s B where D e f i n i t i o n 2.7) Then cat Let on in | B left B T k+l n k + 1 (X) the proof •k+l (X) triangle BX 1 1 T of Theorem Clearly is K + 1 ( B X ) 2.3, the homotopic j is right inclusquare commutative 14 BA = B ( j Also, h o Taking (BA) is o (J)) = just the we have cjT = l o B(J> , (Bj) o diagonal jo<}> = j o J £ o B c } > = map establishes Next namely L-S the in itself, an easy we Ganea category. f to is ( x , l) w i t h of f is C f point * , This its define let the &(0) = Y ^ Let describe tower. because way of shall BX < k Furthermore I n C/^, fibre cat B be define dual . : F = f(x) , C X * and (T. . and an arbitrary f of an Y , 1 £(1) CX a sequence on X, a criterion to determi construction Y be f F structure construction X space 9 another "cocategory" f k+l n BX : BX = A provides this A hoBjoB(J> h o B ( j o c } > ) - h o B A which (Bcf>) C will interesting provides us Ganea). arbitrary consisting = * , f is are topological fibrations, and map. of the The points cofibre functorial space with on f. base 15 k ' 'k 3 as f o l l o w s . where p Q PB k u i Q d e f i n i t i o n of ^ (1) the i n c l u s i o n . goes k + 1 Assuming as C. i ik = E n k F -J<-^ i s the 3* d e f i n e d , the k p E i s the i n c l u s i o n (2) Extend p k k _ k of kJ K C F 7 ^ r k + ] k K B map. to i . , we k k Yk K n k from * , loop follows. Taking the c o f i b r e C PB — ^ fiB every path i n t o i t s endpoint, s p a c e and j' —^ i s the space of a l l path i n B emanating sends where fiB i s the standard f i b r a t i o n : C. 3 get B , 16 k+l F. by mapping the (3) reduced Taking 'k E, ^k cone CF^ to the _ k+1 fibre \+l p F 1 a diagram whose of 1 k+l — —. line is r k + 1 p - k top + * k+l E C have k ~ k+l F we F B the B B B ^ k + i to be defined 17 Expli ci t l y , E P h k + k 1 : is k + = ] the map i C * (t) r k + equivalence 1 (x) is 1 for we C. by x B P k given all satisfying Hence Tower e given k+l = x { ( x , I) have t p -j k + + 1 (x 1 by e | r k 1 h I k 1 (x) = , a) = £ ( 1 ) h. ( x ) k "' o h + = r inductively = k + -j a p k defined k this is the Ganea tower. the map with homotopy- the B 1 3. , . i E and (x , £ ) ' x' : 5* , x is k 1(0)} following 18 Using the functorial property of F. and C. K 1 we k obtain LEMMA i.e. given f 2.9: Ganea's : A —— B in construction C J *, there is is functorial, a commutative di agram n A,n p A,n "A,n F A,n ~n f ^ . n r for all n > 0 about (i) and the smash homotopy, we B,n • — B . Before properties E proving the Since product have the smash BCX next functor = CBX the we need some basic [10]. and B A ( I A X ) satisfies BCX - theorem, C B X = I A ( B A X ) associative (natural homotopy law up , to equivalence). 19 (ii) For locally compact B, there is a natural x e X , b e B ; homeomorphism BX, Y s f wi t h where of F(X,Y) X into map as Y, base is with of f : it = space of so B- has preserves X —— is all compact-open Y as the the continuous topology, mapping pushouts. a pushout CX then f(bx) based and the maps constant point. Since adjoint, Y) r- f(x)(b) the the F X , F(B , diagram If functor we diagram as its right consider the cofibre 20 Bf BX BY Bi 3j Bu BCX i.e. BC f - C R f THEOREM 2.10: space, then BC there is Let B be a natural each n > 0 , such that the ^ ~ B E x,n following x ,n Bp — diagram B X BX F BX,n ^ X . n ^ F Proof: ^BX , n F E By pointed BX,n Bi BF compact map BE. for a locally BX,n induction. ^ ^ R Y B X is commutative 21 (A) When n = Bi 0 , we have x,0 B P ~~ BftX a diagram x,0 BX BPX BX Q. ( B X ) where i which is and Q to gram the commutativity (B) up B i defined restricted The l X ' ° are by ~ P(BX) actually 6(b£) = subspaces of the Assume £ fa we b X > the have - BX 3 : BX b£(t) Bi x > " ~- BE BftX respectively. built I A , a commutative ->^JU» BX BX BX,n (BX) clear. BP. x, n - = to BF. x ,n I £ (t) f a and is ° map with BPX diagram that P dia- , 22 Then we have Bi BF.. „ x ,n X ^ B BF B "L" Bi — X - i i L - BE BX , n« -BX^JU-j: where the upper F the maps middle Bi and x jn n The map i -i B- BX,n+l Y is operating x,n are BX BX,n on * C, x, n rows B A ) M <j> id h l ^ c . is R BX x ,n f B X , n_ lower id C(i) - ^ I U E and "R" F 'Bi n row x ,n tj> f x ,n C r> v ' D v x ,n j n F * J L Q H BX ' id x ,n Br i J ,rwBC. x A ) II id (*) R E x,n+l x ,n obtained by taking cofibres of respectively, defined by considering the diagram 23 where u : CF x ,n 1 x ,n and u are inclusions a : BCF t e I , x, n in — C B F b e B , commutative, the usual is x ,n x e F i.e. : CBF v n . a o Bi = x ,n C way. The defined Then i . by Bi x ,n homotopy a(btx) clearly the equivalence = tbx left with square is 24 The diagram there front chasing, exists As we that back have for squares are pushouts. u~ o a o B i = j ~ o B i a u n i q u e ty , such that commutativity r and and o u = * , of r " L " is "R" , By , hence commutative. notice that o j = Bp, clearly B B r r x , n + 1 x,n + 1 o Bu = B 0 B J'x,n " r x,n l + 8 0 u [ x,n+l r 0 J = x,n Bp x ,n Consequently r o cf, o B J X j n o B i X j n = r - Bp B o j o r v o Bi n x,n B i ^ + 1 0 B j 'x,n S i mi 1 a r l y r o < j ) o B u o B i = r o u o a o B i = * o B i 0 B i x,n 25 and o Bu o B i = * Br„ x , n+1 Hence by the pushout-property, From ( * ) , we get Br BC. r o Bi = o <J> = a commutative x ,n + 1 BX x ,n C ( i ) o cf> BX I I' C. r B X , n + 1^ BX ""BX.n This produces a commutative BE diagram BX x ,n + T 31/ Bq Bp (BX) BX,n+l B C i V n Br x,n + l BX BX C i B X , n r B X , n ^ BX Br * x n + ^ diagram 26 where the the back front square square is x ,n + l p(£) is defined property, B- The = £(0) in there the (A). exists pullback operating map (similarly as is p for : p") , Again a X on — and by unique the X pullback is the : BE 6 n + ^ — — Changing r x,n+l : i C x ,n and r to their homotopy • r • i BX,n+l equivalent p x,n+l : E BX BX,n fibrations x,n+1 X and 'BX,n+l respectively, we have ' c BX,n+l BX by : BX chasing X of defined map diagram i BX , n + 1_ of I — * - 1 and E (BX) pullbackR X . 27 BE Bp x ,n+l BX,n+l p x ,n + l BX BX,n+l BX BX which p is homotopy BX,n+l 1 0 commutative, " R YB Xn, n + l o q o r + Br Since to p make g x n the + -j is plete the x,n+l 0 i B q to = BX,n+l r * B p a f i b r a t i o n , we square Finally 1 due strictly we following use the fact that C(i) o Bq o (J) x,n+1 can change i by i" n + ^ commutative. standard commutative kernel diagram. arguments to com- 28 BF _ J V n ± L _ x,n+l BE B x,n+l x , n +1 _ B X BX n+1 n+1 BX,n+l P THEOREM then B X ,n + l BX 'BX,n + l BX,n+l CW-complex, p 2.11 (T. Ganea): cat X £ k iff Let X be has 3*^ a a based connected cross-section [3]. As tive proof a corollary of "cat COROLLARY connected. of B X < cat 2.12: Furthermore S v X : X — E assume v BX Suppose x, s u c h X , K X" Let cat Proof: theorem that we get be i n TAJ* an alterna- . B,X that < cat cat 2.10, B is X , locally both being compact, then . X £ k < °° , then there exists 29 x,k By • X theorem 2.10, Px,k there 0 S is x = ] x a commutative diagram Bp x , k BE BX x ,k BS. BX E Define t h e PBX,k n 0 S S BX - BX,k R X BX : BX PBX.k 0 E 1 k 0 cat REMARK that if we B is 2.13: locally construct In compact the map fact in B BX,k S x - b B y S Px,k BX ~ 0 B S x i 'k k - u 1 o BS " k J BX ' BX < k we can theorem <j> : B C ^ n drop 2.10 the and s- C R l assumption corrollary 2.12, explicitly 30 along Puppe's proof of the line of we X, rise is the by : the x rule and n X = v(* , • • * , * , that B preserves ~§ by the to the rest chapter, this and f : X —— Y and f = X V X V•••VX , x , the 1 Theorem "structure" (X) gives 4>~ = I where Similarly in map F a natural we shall : BX — E R folding map * • • • * ) = 2.15: V x n : n be in f v 7* discuss co- . We shall f v ••• v f X — 3 —X is defined by . A c o m u l t i p i i c a t i o n on diagram X - X V X X x X X transformation. nilpotency. X,Y The is + of structure-map i k k the tedious. proof T rule the then a certain a structure k of the functors. that but more <{> : X of shown rise in ~ B S , where DEFINITION such that, 18), be T = their (n-copies). a that have gives Let denote "R" w i l l structure-map we k In groups of transformation 2.12 X —=— E show the Hilfssatz Notice structure-map a natural Corollary s 2.14: actually namely, to [7], commutativity REMARK 2.8, (Ref. X is a map 31 is homotopy j is the commutative. i n c l u s i o n of DEFINITION Here the A axis is into the the diagonal Cartesian 2.16: A co-H space X < 1 if is a space map, and product. with a comul t i p i i c a t i o n . Since then as cat a corollary of Theorem THEOREM 2 . 1 7 : homotopy and If DEFINITION 2.18: associative, if X only if we have 2.8, X is a co-H space, is homotopy commutative. a co-H then A comultipiication a the a square XV X a XV X X is xv xv x on so X space, is is BX. DEFINITION space is X) is called X —2-— XvX homotopic to of ^ * The us to and k Let n 2 if each X (of o Ba X with inverse of with n is a system homotopy = Z X v ZY be BX T smash [10]. n (BX) (X, a, associative a co-group. Ba functor This will we shall show the Bx following BX via a natural B X V BX BX properties (1) and be property Define B ( X V X) co- . and x = co-H ° >• X v X A co-group B( X) x) the composite . (X, a, k X — ^ . property Z(XvY) : X homotopy identify homeomorphism a = X 2.20: following frequently, enables and : X — a T inverse, X a co-H space multiplication used A map homotopy DEFINITION consisting 2.19: (2). x) (1) T is In fact, a homotopy by inverse hypothesis, we of have V o C l V - O o c r ^ * and V O ( T V 1 ) O O ~ * A Then from the commutative B X diagram 1 °—=~BXV BX R y V — BX V BX B(l y B ( X we have V o 0 B X V x) x) x V T ) *-—s» B(X V o a = BV o B(1 V X o V =: B * = similarly, V o (x v l B X ) o X) a ( 1 * x x) V T ) a (2) By is homotopy hypothesis associative. (l Vcr) o o - v (oVl A Then from the commutative a BX and a s i m i l a r one, (1 B X diagram BX ^ B X v BX B(XV V we o) X) B(l s- BXVBXVBX * V a) ^-B(XVXVX) have k o a 3 o B ( l v a) = k 3 - k 3 o B (a V 1 ) = k 3 o B(aV1 = we have o Ba A o B (l Va) x (a"V 1 R X o a x ) o a ) o Ba A Consequently ) A o a o 35 THEOREM spaces. (BX, see a", If . T) Remark 2.21: (X, a, x) is Again we have basic B,X a xvx co-commutator X, the then preservation 2.22: co-commutator X of pointed co-group, Let topological so of map i> ( X , a, x ) is 2 2 The be is structure; 2.14. DEFINITION The Let ^> X map ty co-commutator 1 2 v —*- * of weight 1 x map ^ n + °f 1 a co-group. composition x v X 2 2 the be 2 X v X 2 2 is the weight X identity n+1 is map the compos i t i o n x - i DEFINITION of map a xvx 2.23: co-group is <P nul1homotopic integer 1 n + 1 n S the * exists, least we put ^£J!D. The x v x n nilpotency integer rel. nil - n £ 0 base (X, a, n + 1 x nil for points. x) = °° . . (X, a, x) which the If such no Suppose, a co-group. (BX, a , x) Let . It Suppose d i agram as 1j7 2 is i n Theorem be easy now we the to 2 . 2 1 , that basic see have co-commutator that ^ n ( X , a , x) = ty = z k n 0 B ^ k n 2 map o BiJ; i n t 2 n e of 37 Then the diagram Bit? BX B(XV X) B( 1 x V i> ) Vi B(XV X) BXVBX lny\/(B* ) — ^ 2-BXVB( X) n t n id t f BXVBX is commutative Vn and we 1 = ] BX V *n 0 we have shown BX V k n o finite n k o o Bty 6 1 V i> x n v * k n l ° + inductively B *n 1 + that i> n (n = 1 is = n J arbitrary V (BX) have = Thus — — — B X trivial) o k n . B f o r n 38 THEOREM 2 . 2 4 : spaces, and let (X, a, nil nil r reduced is defined by x) k ^. n+1 EX a is be the a be < nil that, is for t]) = n + k 1 . = * o B* = * n+1 a pointed Its space X, the comultipiication 2t], inverse T([X, *) 0 < t < \ (* , [ x , homotopy x) EX V EX ([x, its - ^ a co-group. : EX Then (X, a, =^> topological formula o([x, and pointed co-group. o Bip _,, n+1 well-known suspension B,X ( X , a , x ) = n \p = n+l It x) (BX, a, Proof: Hence Let is t]) defined = [x, 2t-l]) \ < t < 1 by 1-t] 0 < t < 1 39 For conilpotency arbitrary we have, space X , we now define its by conil then pointed as an X = nil immediate coni 1 (EX, a, x) consequence BX < com* 1 X . , of Theorem 2.24 40 Chapter THE FUNCTOR A 3 ~ AND SOME x GENERALIZATIONS A Let consider Then we A the does two pointed topological union with spaces, {+} as base point). have last not category, x X / A strong + = A - , under the it A + x X have weak follows Ax - / 'A for shown and that x * + that the = A X . + smash functor B- Lusternik-Schnirelmann co-ni1 potency . these are also Since not increased . such 2.12 3.1: x X V A category, category Corollary THEOREM /{+} we functor However of = chapter increase Ax - / 'A result X be = A ( d i s j o i n t + A In A and as Let a functor we can improve the follows. A and X be connected pointed AxX / CW-complexes with A locally compact, then cat X = cat /A . 41 Proof: We n e e d to show that c a t X < k ^=%> c a t The s* " part let has i and 7T with TI o i = been shown A X X in : X — ^ Y —s*- X 1 /A < k . corollary be be induced By lemma by 2.9 induced projection we 2.12. have a by To show inclusion respectively, commutative d i agram x, k x, k 'y.k •y,k 7T TT x, k "x , k Suppose P i X —*to there o S E 1 . theorem . exists , y The 2.11. a section then clearly theorem Si J is S^ S„ x now : Y Tr k E o S y established y,k with o i is by an a section appeal 42 COROLLARY as A in theorem-3.1, * /A 1, X 3.2: X is Under the same a co-H space if assumption and only on A , X if . Before proving the next theorem, we need some facts . (a) o , and their let sum Let fi , f f i + f X be a co-H space : Y 2 : 2 X — — X — — X Upon applying In fact, since X — with ^ > 0 a homology (fi+ f )* 2 X is Y be < functor = fi* + f * f 1 = < p1 a 1 » ° ^ = 1 maps, ' f z £ H to then Y we can form . this map, : H(X) 2 a co-H space, XVX P i two f i + f 2 = <f 1 . f z> cr by XVX with comultipiication we , have . have , X H(Y) we X — p 2 ^ a = l XVX P ^ y 0 ' 1 ^ x Then 1 with a* = p. j = 1 or : H(X) — ^ H ( X V X ) 2 . Hence a* On t h e we other hand, from for (x) = the = any H(X) © H(X) x e H (X), (x , x) - ^ ~ H ( X ) we have . diagram have fi* (x) = <f: , f >* i i * (x) = <fi , f >* (x,0) for (x) = <fi , f >* i = <fi , f >* (0,x) . 2 2 Similarly fz* 2 2 * (x) 2 x e H(X). 44 Consequently (fi (b) 0-connected and + f )* 2 Let (x) A and i is cone, hence the , f >* a* = <fi ,f >* (x , x) = <fi , f >* (x , 0 ) = f X be i inclusion A x and 2 * 2 2 (x) pointed X 1-connected. A — 1 — where = <fi + C^ ^ is By V a n 2 * + <f! (x) C. the the , f 2 > * (0 , x) . CW-complexes Consider X f (x) with sequence of A maps, , reduced Kampen's mapping theorem, we have 45 (AxXy ) 'A TT l {i* TTi (A) x e {^(A)} Clearly is pathwise DEFINITION comultipiication f : X — s » x' is a called If a' X and f V f x' We a r e now commutative ready to X hence are a homomorphism or XVX homotopy [6] prove g > x ' v x ' it is 1-connected co-H spaces respectively, square is e J N connected, 3.3: and = *i(A)} with a map primitive if the 46 THEOREM 3.4: with A O-connected X is a co-H space. and g is natural X be : A x w.r.t. X there is /A XV(AX) * the pointed CW-complexes Suppose further a homotopy homomorphism Consider Proof: A and X 1-connected. Then g and Let that equivalence , f : X projections Ax <AX Since is a co-H space, AxX/ a induces This g = induces p + g = a homology g* = (p + q ) * there ^ AxX/ A (pv q)a : A x A is v X /A a comultipiication AxX/ A ^ XV (AX) homomorphism, : H 'AxX, ^ H X V (AX) . 47 Since (p + q ) * = that g* AX is the free by The a is an P* + q * product of naturality follows isomorphism. 2-connected theorem it » [ 1 0 ] , and of TT i (X) J.H.C. of Now TTI = e Ax X V the Klinneth formula X / A i si s 1t -rciovninael c, t e d b ,e i n g (AX) and Whitehead, g w.r.t. from TTI(AX) g is a = e . Therefore homotopy h o m o m o r p h i sms f : equivalence X — » - x' is B trivial . In the rest generalizations of functor which on * 1) of the this smash has F preserves chapter functor we discuss some B- . F the following null objects, Let abstract be properties. null maps and homotopi es. 2) Let T (A) AiX be = n ••• the u = FAiX xA fat {(ai , a | at n wedge {Fpi , ••• ••• xFA n characterized the diagrams , •.• , a ) 2 n least and , Fp } n one a^ = e *} let : F(A x x ••• u> xA ) ^ n . The map is by the commutativity of any 48 We r e q u i r e that i|> : —~ F(T A) n to T n induce (FA) , so that the diagram i> >(A)]- n xA ). F(AiX is T (FA) FAiX n commutative, where ••• j xFA, and j are inclusions. If same A.j = X for argument as all in i , then theorem by 2.8, property we have 2), and using the 49 THEOREM Next, in ing addition 3.5 If assume then 7 that F preserves commutative X e C ! , F satisfies cokernels. k+l ( x ) ^ T K + T h e n we n x is due q~ J = * . to q~ the Under proof is Let 3 * (k+l) are ( F X ) the follow- q F^ these circumstances analogous G be is If to (k+l) canonical fact THEOREM 3 . 6 : The the that FX V <j> have and k+l n x i and 2), X ( F X ) 1 CO k+l q and FX < c a t diagram: T where 1) cat the cokernel X e J * the a functor 1. G preserves null 2. G preserves finite proof on maps. we , C7* , objects, existence of F^ , of and have then of The wcat theorem such null coproducts , FX < w c a t 2.3. X | that maps and i.e. G is homotopies; additive Then as in theorem 2.24, THEOREM 3 . 7 : we If conil have X is a co-group GX < c o n i l X . in , then Chapter 4 COUNTEREXAMPLES In this of our that some more general chapter we previous present results hypotheses, and some are that examples no some longer results to show true are under not dualizable. First (A , X are homology of both we note the i n U J * ) always structure. In f a c t , space have A the consider X X / A same the and XVAX additive short sequence maps: A — with The that i the reduced ^ inclusion homology A x map, x X /A , projection, splits H (AxX) — ^ n P* p the sequence ~ H (A) ^ A X and pi = 1^ accordingly, ~ a ~ H ( A x X , A) - 2 - H _ n n ] (A) 52 giving H ( A x X) = H ( A ) © H ( A x X , A) n n n Now Ax X A = H A x X , A for so Ax X , A = H all n > 0 that Ax X . 1 ^ H (Ax X ) / I AJ " " /H (A) 7 n for On the other hand, from H ( A x X) n we the = H (A) n all n > 0 formula e H (X) n © H (AX) n , have H„(XV AX) = H ( X ) © H (AX) n = H n ( A x X j / . n for all n > 0 [10], 53 Hence the two spaces in groups. Consequently also same. the H space" in their 3.4 1-connected space," then necessarily homotopy EXAMPLE respectively For S 1 , we as the if A xX groups condition to and A homology cohomology the weakened same "X i s "X i s an X v AX Take rings f have A = S (with and 1 Z as H^S ) = Z H (S ) = Z H (S ) = 0 1 1 n are not X = CP coefficient 1 1 n > 1 wi t h as generators e H^S ) e 1 e 1 H^S ) 1 and e°ue° co- arbitrary follows: e° a are equivalent 4.1: cohomology is have additive Nevertheless, theorem singular question = e° , e ^ e 1 = e ^ e and e w 1 e 1 = 0 . 0 = e 1 , . Their ring) are 2 54 For CP , 2 we truncated have at that degree H*(CP ) is 2 3, w i t h the ring y e H (CP ) 2 of as 2 polynomials a polynomial generator. Let structure of us now the compare the multiplicative cohomology space S Yi — x 1 CP / / c i 2 and Y Using the Kunneth 2 = CP formula (S^P ) noting 0 = = . 2 and Tor[Z,0] we V 2 that Tor[Z,Z] have H (Y i ) n hence = © i=0 H^S ) H°(Yi Z H 0 (Yi 1 H (Y! H ° ( S H (Y i H H*(Yi Z H (Yi H M S 2 3 5 ® H 1 1 1 ( S 1 ) 1 ) n _ (CP )/H (S ) i 2 1 H (CP ) - H (CP ) = Z H"(CP ) - 2 ) n 2 2 2 2 Z Z , The generators e e ° ® Y > (e° On the 1 of H (Yi) ® y and ® y)^{e e hand, for ^ ( S ' C P = Y 2 ) H (Yi) and 3 ® y 1 ® y) 1 other , 2 , = we " 5 " ^ H ^ S 1 ) 1 ) are x respectively, 2 ( e ^ e 2 H (Y ) with ® ( Y ^ Y ) = e 1 ® y : have X C 8 P > 2 H N ( C P 2 ) and CP v 2 (S^P ) H (CP ) 2 n © H (S CP ) 2 n (Direct Again by computation, H°(Y ) Z H 0 2 (Y ) 1 2 we H H (Y ) 0 © HMS^P ) H" (Y ) z H (Y ) 0 © H (S'CP ) = 0 2 3 2 , 5 2 2 2 sum o f rings) get H (Y ) 2 1 (CP ) © 0 = Z 2 2 5 2 = 0 H^S ) 1 H (S 1 1 ® H (CP ) = Z ) ® H (CP ) = Z 2 4 2 2 56 But in this H (Y ) are 5 2 e 1 ® y 2 e 1 ® 2 Y thus case, 2 5 1 Yi and are not homotopy of the if a co-H X is X is be H (Y ) , 2 in 3 y^ie with and 2 and 2 space, AVX —P—-s* A The 2 have ® y) 1 = 0 + the then dual answer different 3.2, is, arises A —^—3so is fibre in Ax X . the question, with cohomology cofibre i.e. F. F = of consider Is general, F an 4.2: (EA Notice x that x *)\J{MK X) C EA EA = { £ e A contract!" bl e , so 1 I £(!)=*} that / QA x as the claims X that i . Now the fibra- H-space negative. with is 3.2 in EXAMPLE rings, equivalent. cofibration to one? Y corollary turn seen which H (Y ) 2 0 y e H ^ S ^ P ) 1 respectively 2 So cofibre tion e of • they us 2 H (S CP ) £ generators y e H (CP ) , Next, let the This when can 57 Take and X = S which 3 K ( Z , 2) , A = according to so theorem Consider A — with is cofibres that QA = S 1 3.4. But 3 K v s^ / A _ Z (A dual F of the =- A v above P A x —*w not implication A and We t h e n x X. ) 1 The is an H-space A — 2 — A x EX and respectively. A X Z X = s , cofibrations — K K and K' H-space the A x X — an A K' have ZK is ~ F - A v A QX — ^ - A P 58 with But fibres this F is and false EXAMPLE f and F ^ ftF J A F (even if 4.3: x M we X is Taking / an we , 2) K(Z / 1 have H-space!) A = ^ s'x s QF because Finally respectively, , F fiF as we . see in = X , 2 is an H-space. provide an example to show that in general , cat EXAMPLE the Moore B - ZK' (Z type of 4.4: polyhedra , 1) , BX £ M i n ( c a t Take with X = ZK suspensions (and B = gcd F (Z in B, cat , 1) ; , 1 they particular, . , 2) K (Z (m , n ) = X) . X = the B = Cat B = 1 and cat , are homotopy 1-connected) . Hence cat , 2) Since have they K' (Z X = Cat X = 1 59 Now we consider the short sequence B V X >—-—* B x X »BX 9 have H (B V X ) = H (B) 0 H (X)=0 k k k + 2 ,0 k H ( B V X) = Z 0 H ( B V x) = Z 2 On the other hand, it Tor(Z m © Z n . is well-known m m that , 1) = Z , . = 0 n' (m , n ) and Z By the Kunneth m ® Z formula, n = 1, x (m , n ) = 0 we g e t H (B x x ) = 0 H (B x x) = Z k 0 k + 2 , 0 and H (B 2 x X) = Z m ® Z © Z ® Z n = Z m © Z n . 60 Therefore, which i is a homotopy Hence BX - * , implies cat BX = 0 1 M i n ( c a t B , REMARK 4 . 5 : we equivalence. In g e n e r a l , cat for X) . a pointed space X, have (1) wcat X < cat (2) conil (X) (where with For completeness, strict EXAMPLE moving TTI(X) 7TiX an =f is open and e X is also (X) (1). 4.6: Let 3-cell from is free, not and [11] a Hausdorff base collect in X , two X be the point). examples space a Poincare we space have to obtained 3-sphere. cat show X > 1; by the re- Since (cat X = 1 free). On ^•(X) < wcat nondegenerate we inequalities X < Cat 0 the for other all hand, i , we get X^ ^ 2 H, is (2) 1-connected. = 0 Hence Since (2) 61 is contract!'bl e , consequently q A * * : X — X x X X q ( 2 ) i.e. wcat EXAMPLE iff X is type of a co-H 4.7: It space, and X = is 1 . well Cat known X < 1 iff that X cat has X < 1 the homotopy suspension. T a k i ng X = s where is a a co-H is of order space, cat but p it 3 ^ a with is X < 1 , not but e 2 p p + 1 an , odd prime, a suspension. Cat X > 1 then X Hence [1] 62 REFERENCES [1] I . B e r s t e i n and P . J . H i l t o n , Hopf i n v a r i a n t s , I l l i n o i s J . 451 . [2] I. Berstein Illinois J . [3] T. Ganea, L u s t e r n i k - S c h n i r e l m a n n category c a t e g o r y , I l l i n o i s J . M a t h . , V o l . 11, No. pp. 417-427. [4] P . J . H i l t o n , An I n t r o d u c t i o n t o H o m o t o p y T h e o r y , C a m b r i d g e T r a c t s i n M a t h , and M a t h e m a t i c a l Physics, N o . 43 ( 1 9 5 3 ) . [5] P . J . H i l t o n , Homotopy Breach (1965). [6] P . J . H i l t o n , A l g e b r a i c T o p o l o g y , An I n t r o d u c t o r y Course, Lecture Notes, Courant I n s t i t u t e of Mathematical Sciences (1969). [7] D. P u p p e , H o m o t o p i e m e n g e n und Abbildungen, I. Mathematische pp. 299-344 (1958). and T . G a n e a , Math. (1961), Homotopical pp. 99-130. Theory and generalized pp. 437- nilpotency, Duality, and s t r o n g 3 (1967), Gordon ihre induzierten Z e i t s c h n i f t , Bd. [8] E.H. Spanier, [9] A . H . W a l l a c e , A l g e b r a i c T o p o l o g y , Homology cohomology, W.A. Benjamin Inc. (1970). [10] Algebraic C a t e g o r y and Math. (1962), Topology, McGraw-Hill and 69, (1966). and G.W. W h i t e h e a d , G e n e r a l i z e d Homology T h e o r i e s , Amer. Math. S o c , 1 02 ( 1 9 6 2 ) , p p . 2 2 7 - 2 8 3 . Trans. 63 [11] T . G a n e a , P . J . H i l t o n a n d F . P . P e t e s o n , On t h e h o m o t o p y - c o m m u t a t i v i t y of loop-spaces and suspensions, T o p o l o g y 1 (1962) , pp. 133-161 .
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On numerical homotopy invariants and homotopy functors Chen, Dien Wen 1972
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Title | On numerical homotopy invariants and homotopy functors |
Creator |
Chen, Dien Wen |
Publisher | University of British Columbia |
Date Issued | 1972 |
Description | The main object of study in this paper is the "Smash Functor" BΛ-, which associates to a space X the smash product BX = BΛX. We find that various numerical homotopy invariants, such as strong category, weak category, Lusternik-Schnirelmann category, and in the case where X is a co-group, the nil-potency do not increase under the smash functor, i.e. we have CatBX ≤ CatX, catBX ≤ catX, WcatBX ≤ WcatX and nilBX ≤ nilX. We then consider the particular case of the smash functor where B = A⁺ (disjoint union of A with a point) in which case BX = [sup AxX]/A. This functor actually preserves L-S category. Furthermore we show that (in the category of based spaces of the based homotopy type of CW-complexes) when X is a co-H space, then the spaces [sup AxX]/A and XV(AX) are homotopy equivalent; this is a generalization of [sup A x ΣB]/A ≃ ΣBv(AΣB) . We also investigate conditions a functor F has to satisfy in order to have the properties we found for BΛ- . Finally, we collect a few counterexamples to show that the duals of some of our results are false. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080355 |
URI | http://hdl.handle.net/2429/32442 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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