UBC Theses and Dissertations

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UBC Theses and Dissertations

On numerical homotopy invariants and homotopy functors Chen, Dien Wen 1972

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1\ ^ ON NUMERICAL HOMOTOPY INVARIANTS AND HOMOTOPY FUNCTORS by D I E N WEN CHEN B.Sc., N a t i o n a l Taiwan U n i v e r s i t y , 1963 M.Sc., N a t i o n a l Tsin-Hua U n i v e r s i t y , 1966 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF P H I L O S O P H Y i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE U N I V E R S I T Y OF B R I T I S H C O L U M B I A M a r c h , 1 9 7 2 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r -s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f MATHEMATICS T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e S u p e r v i s o r : D r . A r m i n F r e i A b s t r a c t The m a i n o b j e c t o f s t u d y i n t h i s p a p e r i s t h e " S m a s h F u n c t o r " B A - , w h i c h a s s o c i a t e s t o a s p a c e X t h e s m a s h p r o d u c t BX = B A X . We f i n d t h a t v a r i o u s n u m e r i c a l h o m o t o p y i n v a r i a n t s , s u c h as s t r o n g c a t e g o r y , weak c a t e g o r y , L u s t e r n i k - S c h n i r e l m a n n c a t e g o r y , a n d i n t h e c a s e w h e r e X i s a c o - g r o u p , t h e n i l -p o t e n c y do n o t i n c r e a s e u n d e r t h e s m a s h f u n c t o r , i . e . we h a v e C a t B X < C a t X , c a t B X < c a t X , W c a t B X < W c a t X a n d n i l B X < n i l X . We t h e n c o n s i d e r t h e p a r t i c u l a r c a s e o f t h e s m a s h f u n c t o r w h e r e B = A + ( d i s j o i n t u n i o n o f A w i t h a p o i n t ) i n w h i c h c a s e BX = A x X / A m T h i s f u n c t o r a c t u a l l y p r e s e r v e s L - S c a t e g o r y . F u r t h e r m o r e we show t h a t ( i n t h e c a t e g o r y o f b a s e d s p a c e s o f t h e b a s e d h o m o t o p y t y p e o f C W - c o m p l e x e s ) when X i s a c o - H s p a c e , t h e n t h e s p a c e s A * X / A a n d X V ( A X ) a r e h o m o t o p y e q u i v a l e n t ; t h i s i s a g e n e r a l i z a t i o n o f = Z B v ( A Z B ) . We a l s o i n v e s t i g a t e c o n d i t i o n s a f u n c t o r F h a s t o s a t i s f y i n o r d e r t o h a v e t h e p r o p e r t i e s we f o u n d f o r B A - . F i n a l l y , we c o l l e c t a few c o u n t e r e x a m p l e s t o show t h a t t h e d u a l s o f some o f o u r r e s u l t s a r e f a l s e . i i T A B L E OF CONTENTS P a g e A B S T R A C T i i ACKNOWLEDGEMENTS i v C h a p t e r 1 I N T R O D U C T I O N AND NOTATION 1 2 THE SMASH FUNCTOR 4 3 THE FUNCTOR A ^ ~ AND SOME G E N E R A L I Z A T I O N S . . . 40 4 C O U N T E R E X A M P L E S 51 R E F E R E N C E S 62 i i i ACKNOWLEDGEMENTS T h e a u t h o r i s d e e p l y i n d e b t e d t o P r o f e s s o r A . F r e i f o r h i s i n v a l u a b l e a s s i s t a n c e a n d e n c o u r a g e m e n t . He i s a l s o g r a t e f u l f o r t h e f i n a n c i a l a s s i s t a n c e o f t h e M a t h e m a t i c s D e p a r t m e n t o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . i v 1 C h a p t e r 1 INTRODUCTION AND NOTATION T h e m a i n o b j e c t o f s t u d y i n t h i s p a p e r i s t h e " S m a s h F u n c t o r " B A - , w h i c h a s s o c i a t e s t o a s p a c e X t h e s m a s h p r o d u c t BX = BAX . We f i n d t h a t v a r i o u s n u m e r i c a l h o m o t o p y i n v a r i a n t s , s u c h as s t r o n g c a t e g o r y , weak c a t e g o r y , L u s t e r n i k -S c h n i r e l m a n n c a t e g o r y , a n d ( i n t h e c a s e when X i s a c o - g r o u p ) t h e n i l . p o t e n c y , do n o t i n c r e a s e u n d e r t h e s m a s h f u n c t o r , i . e . we h a v e C a t BX < C a t X , c a t BX < c a t X , w c a t BX < w c a t X a n d n i l BX < n i l X . T h e s e r e s u l t s a r e g i v e n i n C h a p t e r 2. I n C h a p t e r 3 we c o n s i d e r a p a r t i c u l a r e x a m p l e o f t h e s m a s h f u n c t o r w h e r e B = A + ( d i s j o i n t u n i o n o f A w i t h Ax X a p o i n t ) i n w h i c h c a s e BX = / ^ . T h i s f u n c t o r a c t u a l l y p r e s e r v e s L - S c a t e g o r y . F u r t h e r m o r e we show t h a t ( i n t h e c a t e g o r y o f b a s e d s p a c e s o f t h e b a s e d h o m o t o p y t y p e o f AxX C W - c o m p l e x e s ) when X i s a c o - H s p a c e , t h e s p a c e / ^ a n d X V ( A X ) a r e h o m o t o p y e q u i v a l e n t ; t h i s i s a g e n e r a l i z a t i o n A x y R o f H i l t o n ' s f o r m u l a / A a Z B V ( A E B ) . I n t h i s c h a p t e r 2 we a l s o i n v e s t i g a t e c o n d i t i o n s a f u n c t o r F h a s t o s a t i s f y i n o r d e r t o h a v e t h e p r o p e r t i e s we f o u n d f o r BA- . I n C h a p t e r 4 , we c o l l e c t a f e w c o u n t e r e x a m p l e s t o show t h a t t h e d u a l s o f some o f o u r r e s u l t s a r e f a l s e . T h r o u g h o u t t h i s p a p e r we s h a l l w o r k i n t h e c a t e -g o r y o f b a s e d s p a c e s h a v i n g t h e b a s e d h o m o t o p y t y p e o f CW-c o m p l e x e s . U n l e s s o t h e r w i s e s t a t e d , a l l maps a n d h o m o t o p i e s a r e s u p p o s e d t o p r e s e r v e b a s e p o i n t , w h i c h i s a l w a y s d e n o t e d b y * . We s h a l l u s e t h e f o l l o w i n g n o t a t i o n s . L e t A a n d B be p o i n t e d t o p o l o g i c a l s p a c e s , t h e n 1) I = [ 0 , 1 ] i s t h e c l o s e d u n i t i n t e r v a l w i t h 0 = * 2) A * i s t h e f r e e p a t h s p a c e o v e r A ( w h e r e we do n o t c o n s i d e r I as a b a s e d s p a c e ) 3) PA i s t h e b a s e d p a t h s p a c e o v e r A , i . e . PA = {ui e A 1 | u ( 0 ) = *} . I t i s a c o n t r a c t i b l e s p a c e . 4 ) ftA i s t h e l o o p s p a c e o v e r A , i . e . ftA = { U E A 1 | u ( 0 ) = o ) ( l ) = *} A V B i s t h e o n e - p o i n t u n i o n o f A a n d B . A B = A X B / A V B = A A B i s t h e s m a s h p r o d u c t o f A a n d B . " = " means h o m e o m o r p h i c a n d " * " means h o m o t o p i c . 4 C h a p t e r 2 THE SMASH FUNCTOR L e t J£ be t h e c a t e g o r y o f p o i n t e d s p a c e s w i t h b a s e p o i n t - p r e s e r v i n g c o n t i n u o u s m a p s , a n d l e t B be a f i x e d s p a c e i n 3£ • T h e n B - i s a f u n c t o r f r o m t o 2 £ . T h a t i s , i t a s s o c i a t e s a s p a c e BX t o a n y s p a c e X e 0 ^ , a n d t o a n y map f : X Y i n 3* , i t a s s o c i a t e s a map B f : BX — - BY s u c h t h a t B l x = 1 B X a n d B ( f o g ) = B f o B g . C l e a r l y t h e f u n c t o r B - p r e s e r v e s n u l l maps a n d we a l s o h a v e LEMMA 2 . 1 : The f u n c t o r B - p r e s e r v e s h o m o t o p y . i . e . i f a * 3 ( r e l * ) : X Y t h e n B a = B 3 ( r e l * ) : BX — * BY . Proof: S i n c e a - 3 ( r e l * ) : X — * Y , t h e r e e x i s t s H : X x I ^ Y w i t h H ( x,0) = a ( x ) H ( x , l ) = 8 ( x ) 5 C o n s i d e r t h e d i a g r a m 1 B x H B x X x I B x Y p x 1 BX x I BY w h e r e p , q a r e i d e n t i f i c a t i o n m a p s . F o r b e B , x e X , t e I , we h a v e q o 1 B x H b , x b , H ( x , t ) bH x , t I f b = * o r x = * , t h e n b H ( x , t ) = * . S i n c e p i s an i d e n t i f i c a t i o n map a n d I i s c o m p a c t , i t f o l l o w s t h a t p x l j i s a l s o an i d e n t i f i c a t i o n m a p . T h e r e f o r e t h e r e i s a map H" : BX x I — 3 - BY s u c h t h a t t h e a b o v e d i a g r a m i s c o m m u t a t i v e . O b v i o u s l y IT i s t h e r e q u i r e d h o m o t o p y o f B a a n d B3 . I F r o m Lemma 2 .1 we c a n s e e t h a t t h e f u n c t o r B -a c t u a l l y o p e r a t e s on The h o m o t o p y c a t e g o r y o f b a s e d t o p o l o g i c a l s p a c e s . I t s o b j e c t s a r e p o i n t e d s p a c e s , i t s 6 m o r p h i s m s a r e h o m o t o p y c l a s s e s o f c o n t i n u o u s m a p s . C l e a r l y B - i s a l s o a f u n c t o r on " U J * , t h e h o m o t o p y c a t e g o r y o f b a s e d h o m o t o p y t y p e o f C W - c o m p l e x e s , p r o v i d e d B e bJ* . c~r k + l L e t X e J+ a n d l e t II X be t h e c a r t e s i a n p r o d u c t i . . . k + l o f k + l c o p i e s o f X , a n d l e t T ( X ) C n X be t h e s u b s p a c e c o n s i s t i n g o f p o i n t s ( x 1 , x 2 , x k + 1 ) s u c h t h a t x . = * f o r a t l e a s t o n e i ( l < i < k + l ) . k + l L e t A : X — » II X be t h e d i a g o n a l m a p , i . e . A ( x ) = ( x , x , • • • • , x ) . L e t k + l x ( k + i ) = / T k + i ( x ) be t h e ( k + l ) - s m a s h e d p o w e r o f X w i t h q : n X x ( k + l ) t h e i d e n t i f i c a t i o n m a p . D E F I N I T I O N 2 . 2 : X h a s w e a k c a t e g o r y < k ( w r i t t e n w c a t X < k ) I f f q A « * . THEOREM 2 . 3 : L e t B , X be p o i n t e d s p a c e s , t h e n w c a t BX < w c a t X . 7 Proof: I f w c a t X be p r o v e d ; s o , l e t us a s s u m e ( a n o n n e g a t i v e i n t e g e r ) . By = 0° , t h e n n o t h i n g n e e d s t o t h a t w c a t x < k w i t h k f i n i t e d e f i n i t i o n 2.2, we h a v e k+l n x 1 (k+l) w i t h q A - * , t h e n Bq o BA = B ( q o A) - B * = * C o n s i d e r BX B A rk+1 N n x _Bg_ (1) k+l n B X x BX x ( k + l ) k I (k+l) w h e r e h i s t h e u n i q u e map d e f i n e d by t h e c o m m u t a t i v e d i a g r a m s : 8 k + 1 > KXnV »v _ -th i n w h i c h p i : n X =- X a n d p i : n BX — BX a r e i p r o j e c t i o n s . T h e map k i s d e f i n e d as t h e c o m p o s i t i o n o f maps : 6 1 X < k + D x ( k + 1 ) , _ E ( k + i ) x ( k + i ) t :m | B X j ( k + 1 ) i n w h i c h t h e map 6 i s t h e c o m p o s i t i o n o f m a p s , A^k+n B ^ J U B ( k + i ) a n d t i s t h e h o m e o m o r p h i s m w h i c h i n t e r c h a n g e s t h e o r d e r o f B ' s a n d X ' s i n t h e s m a s h p r o d u c t . I t i s e a s i l y c h e c k e d t h a t t h e r i g h t s q u a r e i n ( 1 ) i s c o m m u t a t i v e . F u r t h e r m o r e , f o r i = 1 , 2 , , k + l , p / o h o ( B A ) = ( B p . ) o ( B A ) = B ( p i o A) = l g x = p . o A , w h i c h p r o v e s t h a t h o ( B A ) = A . T h u s ( 1 ) i s a c o m m u t a t i v e d i a g r a m . S i n c e q ~ o A = q " o h o ( B A ) = k p B q o B A = k o B ( q o A) = * , we h a v e w c a t BX < k . B 9 D E F I N I T I O N 2 . 4 : The s t r o n g c a t e g o r y C a t X o f a t o p o l o g i c a l s p a c e X i s t h e l e a s t i n t e g e r k > 0 w i t h t h e p r o p e r t y t h a t X h a s t h e h o m o t o p y t y p e o f a C W - c o m p l e x w h i c h may be c o v e r e d by k + l s e l f - c o n t r a c t i b l e s u b c o m p l e x e s . I f no s u c h i n t e g e r e x i s t s , t h e n C a t X = °° . THEOREM 2 . 5 : L e t X e k / * be c o n n e c t e d a n d l e t B be a l o c a l l y f i n i t e C W - c o m p l e x , t h e n C a t BX < C a t X . Proof: S i n c e C a t BX < C a t X i s c e r t a i n l y t r u e f o r C a t X = °° , l e t us a s s u m e t h a t C a t X < k < °° . W i t h -o u t l o s s o f g e n e r a l i t y , we may a s s u m e t h a t X i t s e l f i s a C W - c o m p l e x w h i c h i s c o v e r e d by k + l s e l f - c o n t r a c t i b l e s u b -c o m p l e x e s X m . F u r t h e r m o r e we may a l s o a s s u m e t h a t * b e l o n g s t o e a c h X m , a n d t h a t t h e s e l f - c o n t r a c t i n g h o m o t o p i e s H • X x I — X r e s p e c t * . m r L e t 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 D x H : B x X x I — B x X m , as i n t h e J B m m m p r o o f o f Lemma 2 . 1 , t h e n H"m : 1 B ^ = * ( r e l * ) . k + l m C l e a r l y BX = VJ B X m , h e n c e BX i s c o v e r e d by k + l s e l f -c o n t r a c t i b l e s u b c o m p l e x e s BX . C o n s e q u e n t l y C a t BX < k . I D E F I N I T I O N 2 . 6 : The L u s t e r n i k - S c h n i r e l m a n n c a t e -g o r y , c a t X , o f a t o p o l o g i c a l s p a c e X i s t h e l e a s t i n t e g e r 10 k > 0 w i t h t h e p r o p e r t y t h a t X may be c o v e r e d by k + l o p e n s u b s e t s w h i c h a r e c o n t r a c t i b l e i n X . I f no s u c h i n t e g e r e x i s t s , t h e n c a t X = °° . We s h a l l p r e s e n t an a l t e r n a t i v e d e f i n i t i o n o f L - S c a t e g o r y c a t X , d u e t o G . W . W h i t e h e a d , a n d show t h a t i t i s e q u i v a l e n t t o t h e p r e v i o u s o n e when X i s i n l < J * . F i r s t we r e m a r k t h a t by c l a s s i c a l a r g u m e n t s , i t f o l l o w s t h a t i f X i s a C W - c o m p l e x , t h e n we may r e p l a c e t h e o p e n s u b s e t s i n D e f . 2 . 6 by c l o s e d s u b s e t s o r e v e n by s u b c o m p l e x e s . D E F I N I T I O N 2 . 7 : L e t k+l n X , T I k+l k+l ( X ) C n X be de-I f i n e d as i n D e f i n i t i o n 2 . 2 . i f t h e r e e x i s t s a map <f> : X A = j o <j> , T h e n c a t X < k i f a n d o n l y T k + l 1 ( X ) s u c h t h a t k+l n x i T k + 1 ( X ) w h e r e j i s t h e i n c l u s i o n a n d A i s t h e d i a g o n a l m a p . 11 We do n o t c l a i m t h a t D e f . ' s 2 . 6 a n d 2 . 7 a r e e q u i v -a l e n t t h r o u g h o u t t h e c a t e g o r y CT* , h o w e v e r t h e y a r e e q u i v a -l e n t on t h e p o i n t e d c a t e g o r y o f C W - c o m p l e x e s . I n f a c t l e t k+l i. = { ( x 1 , . . . . x . _ 1 , * , x 1 + 1 , - - - - x k + 1 ) | x j e x , i f j < k + i } c n x , k+l . a n d l e t p : n X *- X be t h e I p r o j e c t i o n , *> I 1 < £ < k + l . T h e n i f t h e r e e x i s t s c}> s u c h t h a t A = j o <J> , we h a v e l x = pz o A « pz o j o <j> . k+l C o n s i d e r c j T M Z . ) . S i n c e T k + 1 ( X ) = \J z . , t h e s e c j > _ 1 ( Z . ) 1 i = l 1 f o r m a c o v e r i n g o f X . F u r t h e r m o r e we h a v e p^ j <P - P^A = l x , w i t h p i j (j) t p ~ 1 ( Z . ) j = * . T h u s X i s c o v e r e d by k + l c l o s e d s u b s e t s w h i c h a r e c o n t r a c t i b l e i n X . S i n c e X i s a C W - c o m p l e x , we h a v e c a t X < k . C o n v e r s e l y , s u p p o s e c a t X < k . T h e n k+l X = W X m m=l w i t h X m a s u b c o m p l e x c o n t r a c t i b l e i n X . T h e n , by t h e h o m o t o p y e x t e n s i o n p r o p e r t y , t h e r e e x i s t s F m : X * I — w h i c h c o n t r a c t s X m F m ( x , 0) = x F m ( x , l ) - * x e X x e X m , m 1 < m < k + l . T h e n t h e s e F m ' s d e t e r m i n e a u n i q u e map - { F l » F 2 ' " * - > F m ' : X x 1 k+l n x i s u c h t h a t c o m m u t e s . Now F ( x,0) = A ( x ) a n d F ( x , l ) e T ( X ) , i . t h e r e e x i s t s <J>(x) == F ( x , l ) s u c h t h a t A - j o <J> . 13 THEOREM 2 . 8 : ( B a s e d on D e f i n i t i o n 2 . 7 ) L e t B , X e . T h e n c a t BX s c a t X . Proof: When c a t X = °° , t h e c o n c l u s i o n i s t r i v i a l . L e t us a s s u m e t h a t c a t X < k < °° , t h e n by d e f i n i t i o n t h e r e e x i s t s <j> , s u c h t h a t j <j> = A . C o n s i d e r t h e f o l l o w i n g d i a g r a m rk+1 B n X I 1 BX ~ B T k + 1 ( X ) k+l n B X 1 T K + 1 ( B X ) w h e r e h i s d e f i n e d as i n t h e p r o o f o f T h e o r e m 2 . 3 , j i s i n c l u -s i o n m a p , a n d £ = h | B •k+l ( X ) C l e a r l y t h e r i g h t s q u a r e i s c o m m u t a t i v e . The l e f t t r i a n g l e i s h o m o t o p i c c o m m u t a t i v e b e c a u s e 14 BA = B ( j o (J)) = ( B j ) o (Bcf>) A l s o , h o ( B A ) i s j u s t t h e d i a g o n a l map A : BX T a k i n g cjT = l o B(J> , we h a v e k + l n BX j o < } > = j o J £ o B c } > = h o B j o B ( J > h o B ( j o c } > ) - h o B A = A w h i c h e s t a b l i s h e s c a t BX < k . 9 N e x t we s h a l l d e s c r i b e a n o t h e r s t r u c t u r e on X , n a m e l y t h e G a n e a t o w e r . T h i s p r o v i d e s a c r i t e r i o n t o d e t e r m i L - S c a t e g o r y . F u r t h e r m o r e t h i s c o n s t r u c t i o n i s i n t e r e s t i n g i n i t s e l f , b e c a u s e i t s d u a l c o n s t r u c t i o n w i l l p r o v i d e s us an e a s y way t o d e f i n e " c o c a t e g o r y " ( T . G a n e a ) . I n C/^ , l e t f : X Y be an a r b i t r a r y m a p . T h e f i b r e o f f i s t h e s p a c e F f C X * Y 1 , c o n s i s t i n g o f p o i n t s ( x , l) w i t h &(0) = f ( x ) , a n d £ ( 1 ) = * , a n d t h e c o f i b r e o f f i s C f = Y ^ CX . F f a n d C f a r e f u n c t o r i a l on f . L e t B be an a r b i t r a r y t o p o l o g i c a l s p a c e w i t h b a s e p o i n t * , d e f i n e a s e q u e n c e o f f i b r a t i o n s , 15 k ' 'k uk as f o l l o w s . 3 i s t h e s t a n d a r d f i b r a t i o n fiB — ^ PB — ^ B , where PB i s t h e s p a c e o f a l l p a t h i n B e m a n a t i n g f r o m * , p Q s e n d s e v e r y p a t h i n t o i t s e n d p o i n t , fiB i s t h e l o o p s p a c e and i Q t h e i n c l u s i o n . A s s u m i n g 3*k d e f i n e d , t h e d e f i n i t i o n o f ^ k + 1 g o e s as f o l l o w s . ( 1 ) T a k i n g t h e c o f i b r e C. o f i . , we g e t i k K C i = E k Y C F k n k K 7 k K n k p k F k - J < - ^ E k _ J ^ B 3 where j ' k i s t h e i n c l u s i o n map. ( 2 ) E x t e n d p k t o r k + ] : C. 16 k+l F. ' k E , ^k B by m a p p i n g t h e r e d u c e d c o n e C F ^ t o * (3) T a k i n g t h e f i b r e F k + 1 o f r k + 1 _ \+l F p k + l p Fk+1 ~ E k + l - B k+l C — B 1 k — . B we h a v e a d i a g r a m w h o s e t o p l i n e i s t h e ^ k + i t o be d e f i n e d 17 E x p l i c i t l y , E k + ] = { ( x , I) e C . x B 1 | r k + 1 ( x ) = 1(0)} , P k + 1 i s t h e map g i v e n by P k + 1 ( x , a) = £ ( 1 ) , a n d t h e map h k : C i k + l i s g i v e n by h . ( x ) = ( x , £ ) w i t h k 1 " ' x ' x ' h k i s a h o m o t o p y -e q u i v a l e n c e s a t i s f y i n g p k + - j o h k = r k + - j . * x ( t ) = r k + 1 ( x ) f o r a l l t e I H e n c e we h a v e i n d u c t i v e l y d e f i n e d t h e f o l l o w i n g T o w e r 1 : 5* i p k E k B 1 3. t h i s i s t h e G a n e a t o w e r . 18 U s i n g t h e f u n c t o r i a l p r o p e r t y o f F. a n d C . we K 1 k o b t a i n LEMMA 2 . 9 : G a n e a ' s c o n s t r u c t i o n i s f u n c t o r i a l , i . e . g i v e n f : A — — B i n C J * , t h e r e i s a c o m m u t a t i v e d i a g r a m n A , n F A , n ~ fn r ^ . n p A , n " A , n E B , n • — B f o r a l l n > 0 . B e f o r e p r o v i n g t h e n e x t t h e o r e m , we n e e d some b a s i c p r o p e r t i e s a b o u t t h e s m a s h f u n c t o r [ 1 0 ] . ( i ) S i n c e B C X = B A ( I A X ) a n d C B X = I A ( B A X ) , a n d t h e s m a s h p r o d u c t s a t i s f i e s t h e a s s o c i a t i v e l a w up t o h o m o t o p y , we h a v e B C X - C B X ( n a t u r a l h o m o t o p y e q u i v a l e n c e ) . 19 ( i i ) F o r l o c a l l y c o m p a c t B , t h e r e i s a n a t u r a l h o m e o m o r p h i s m B X , Y s F X , F ( B , Y) f r-w i t h f ( x ) ( b ) = f ( b x ) x e X , b e B ; w h e r e F ( X , Y ) i s t h e s p a c e o f a l l c o n t i n u o u s b a s e d maps o f X i n t o Y , w i t h t h e c o m p a c t - o p e n t o p o l o g y , a n d t h e c o n s t a n t map as b a s e p o i n t . S i n c e B - h a s t h e m a p p i n g f u n c t o r as i t s r i g h t a d j o i n t , i t p r e s e r v e s p u s h o u t s . I f we c o n s i d e r t h e c o f i b r e o f f : X — — Y as a p u s h o u t d i a g r a m CX t h e n s o i s t h e d i a g r a m 20 BX B i B f BY 3 j BCX Bu BC i . e . B C f - C R f THEOREM 2.10: L e t B be a l o c a l l y c o m p a c t p o i n t e d s p a c e , t h e n t h e r e i s a n a t u r a l map B E . B X , n f o r e a c h n > 0 , s u c h t h a t t h e f o l l o w i n g d i a g r a m i s c o m m u t a t i v e B i Bp BF ^ ~ B E — B X x , n x , n BX F ^ X . n ^ F ^BX , n ^ R Y  F B X , n E B X , n ^ B X Proof: By i n d u c t i o n . 21 ( A ) When n = 0 , we h a v e a d i a g r a m B i BftX x,0 B P x , 0 ~~ BPX Q. BX BX ( B X ) l B X ' ° ~ P ( B X ) P b X > ° - BX w h e r e i Q a n d i a r e a c t u a l l y t h e map 3 : BX I I - ( B X ) A , w h i c h i s d e f i n e d by 6 ( b £ ) = £ f a w i t h £ f a ( t ) = b £ ( t ) , r e s t r i c t e d t o t h e s u b s p a c e s BPX a n d BftX r e s p e c t i v e l y . T h e c o m m u t a t i v i t y o f t h e d i a g r a m i s c l e a r . ( B ) A s s u m e t h a t we h a v e b u i l t a c o m m u t a t i v e d i a -g r a m up t o B F . B i x , n x > " ~- BE B P . x , n ->^JU» BX BX B X , n 22 T h e n we h a v e (*) B i R i B F . . „ X ^ B E B J x , r w B C . A ) II ' B r x , n * J L Q H BX i d B i B F — X - i i L - BE n x , n x , n n i d " L " j f x , n tj> "R" F i d ' B i BX F - B X ^ J U - j : h l ^ c . ' D v « C r> v _ -i f x , n C ( i ) i d B X , n + l BX , n B X , n BX B X , n w h e r e t h e u p p e r row i s B - o p e r a t i n g o n F - ^ I U E x , n x , n * x , n + l x , n C , x , n t h e m i d d l e a n d l o w e r r o w s a r e o b t a i n e d by t a k i n g c o f i b r e s o f maps B i n a n d i R Y r e s p e c t i v e l y , x j n B A ) M T h e map <j> i s d e f i n e d by c o n s i d e r i n g t h e d i a g r a m 23 w h e r e 1 x , n C B i x , n a r e i n c l u s i o n s i n t h e u s u a l w a y . The h o m o t o p y e q u i v a l e n c e a : BCF — C B F i s d e f i n e d by a ( b t x ) = t b x w i t h x , n x , n t e I , b e B , x e F v n . T h e n c l e a r l y t h e l e f t s q u a r e i s c o m m u t a t i v e , i . e . a o B i = i . u : CF x , n a n d u : CBF x , n 24 T h e f r o n t a n d b a c k s q u a r e s a r e p u s h o u t s . By d i a g r a m c h a s i n g , we h a v e u~ o a o B i = j ~ o B i , h e n c e t h e r e e x i s t s a u n i q u e ty , s u c h t h a t " L " i s c o m m u t a t i v e . As f o r c o m m u t a t i v i t y o f " R " , n o t i c e t h a t r o u = * , r o j = Bp, a n d t h a t c l e a r l y B r x , n + 1 o Bu = B r x , n + l 0 u B r x , n + 1 0 B J ' x , n " 8 [ r x , n + l 0 J x , n = Bp x , n C o n s e q u e n t l y r o cf, o B J X j n o B i X j n = r o j o B i ^ - B p v n o B i B r x , n + 1 0 B j ' x , n 0 B i 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 a n d B r „ o Bu o B i = * o B i = * x , n+1 H e n c e by t h e p u s h o u t - p r o p e r t y , r o <J> = B r x n + ^ F r o m ( * ) , we g e t a c o m m u t a t i v e d i a g r a m 25 B r BC. x , n + 1 BX x , n C ( i ) o cf> I ' BX I C . r B X , n + 1 ^ BX " " B X . n T h i s p r o d u c e s a c o m m u t a t i v e d i a g r a m BE x ,n + T BX 3 1 / B X , n + l Bq ( B X ) B C i B r V n x , n + l Bp C i B X , n r B X , n ^ BX BX BX 26 w h e r e t h e f r o n t s q u a r e i s t h e p u l l b a c k o f BX , n + 1_ t h e b a c k s q u a r e i s B - o p e r a t i n g on t h e p u l l b a c k o f x , n + l X i s d e f i n e d by I T h e map p : X — p ( £ ) = £(0) ( s i m i l a r l y f o r p") , a n d t h e map 6 : B X 1 — * - ( B X ) i s d e f i n e d as i n ( A ) . A g a i n by d i a g r a m c h a s i n g a n d p u l l b a c k -p r o p e r t y , t h e r e e x i s t s a u n i q u e i : B E X n + ^ — — E R X . C h a n g i n g r x , n + l : C i x , n a n d r • r B X , n + l • i BX B X , n t o t h e i r h o m o t o p y e q u i v a l e n t f i b r a t i o n s p x , n + l : E x , n + 1 X a n d ' B X , n + l ' c B X , n + l BX r e s p e c t i v e l y , we h a v e 27 BE x , n + l B X , n + l Bp x , n + l p B X , n + l BX BX BX w h i c h i s h o m o t o p y c o m m u t a t i v e , d u e t o t h e f a c t t h a t p B X , n + l 0 1 " r R Y n + 1 o q o i = r B X , n + l B X , n + l C ( i ) o (J) o Bq B r x , n + l 0 B q * B p x , n + 1 S i n c e p g x n + -j i s a f i b r a t i o n , we c a n c h a n g e i by i " n + ^ t o make t h e s q u a r e s t r i c t l y c o m m u t a t i v e . F i n a l l y we u s e s t a n d a r d k e r n e l a r g u m e n t s t o c o m -p l e t e t h e f o l l o w i n g c o m m u t a t i v e d i a g r a m . 28 B F _ J V n ± L _ BE B p x , n + 1 _ B X x , n + l x , n + l n + 1 n + 1 BX B X , n + l B X , n + l ' B X , n + l P B X , n + l B X THEOREM 2 . 1 1 ( T . G a n e a ) : L e t X be a b a s e d c o n n e c t e d C W - c o m p l e x , t h e n c a t X £ k i f f 3*^ has a c r o s s - s e c t i o n [ 3 ] . As a c o r o l l a r y o f t h e o r e m 2 . 1 0 , we g e t an a l t e r n a -t i v e p r o o f o f " c a t B X < c a t X " . COROLLARY 2 . 1 2 : L e t B , X be i n TAJ* , b o t h b e i n g c o n n e c t e d . F u r t h e r m o r e a s s u m e t h a t B i s l o c a l l y c o m p a c t , t h e n c a t B X < c a t X . Proof: S u p p o s e c a t X £ k < °° , t h e n t h e r e e x i s t s S v : X — E v x, s u c h t h a t X X , K 29 x , k X • P x , k 0 S x = ] x By t h e o r e m 2 . 1 0 , t h e r e i s a c o m m u t a t i v e d i a g r a m Bp x , k BE x , k BS. BX BX E B X , k BX D e f i n e S R X : BX E B X , k b y S B X ~ ' k u " J k i k o BS t h e n P B X , k 0 S B X - P B X . k 0 1 k 0 B S x - B P x , k 0 B S x - 1 BX ' c a t BX < k REMARK 2 . 1 3 : I n f a c t we c a n d r o p t h e a s s u m p t i o n t h a t B i s l o c a l l y c o m p a c t i n t h e o r e m 2 . 1 0 a n d c o r r o l l a r y 2 . 1 2 , i f we c o n s t r u c t t h e map <j> : B C ^ n s - C R l - e x p l i c i t l y 30 a l o n g P u p p e ' s l i n e ( R e f . [ 7 ] , H i l f s s a t z 1 8 ) , b u t t h e n t h e p r o o f o f t h e c o m m u t a t i v i t y o f " R " w i l l be m o r e t e d i o u s . REMARK 2 . 1 4 : N o t i c e t h a t , i n t h e p r o o f o f T h e o r e m 2 . 8 , we a c t u a l l y show t h a t B p r e s e r v e s a c e r t a i n " s t r u c t u r e " o f X , n a m e l y , t h e s t r u c t u r e - m a p <{> : X T k + 1 ( X ) g i v e s r i s e t o t h e s t r u c t u r e - m a p ~§ by t h e r u l e 4>~ = w h e r e I i s a n a t u r a l t r a n s f o r m a t i o n o f f u n c t o r s . S i m i l a r l y i n C o r o l l a r y 2 . 1 2 we h a v e s h o w n t h a t t h e s t r u c t u r e - m a p s : X —=— E x k g i v e s r i s e t o a s t r u c t u r e map F : BX — E R X by t h e r u l e T = ~ k B S , w h e r e i k i s a n a t u r a l t r a n s f o r m a t i o n . I n t h e r e s t o f t h i s c h a p t e r , we s h a l l d i s c u s s c o -g r o u p s a n d t h e i r n i l p o t e n c y . L e t X , Y a n d f : X — — Y be i n 7* . We s h a l l d e n o t e n X = X V X V • • • V X , a n d n f = f v f v • • • v f ( n - c o p i e s ) . The f o l d i n g map V : n X — 3 — X i s d e f i n e d by v ( * , • • * , * , x , * • • • * ) = x . D E F I N I T I O N 2 . 1 5 : A c o m u l t i p i i c a t i o n on X i s a map a s u c h t h a t t h e d i a g r a m X - X V X X x X 31 i s h o m o t o p y c o m m u t a t i v e . H e r e A i s t h e d i a g o n a l m a p , a n d j i s t h e i n c l u s i o n o f t h e a x i s i n t o t h e C a r t e s i a n p r o d u c t . D E F I N I T I O N 2 . 1 6 : A c o - H s p a c e i s a s p a c e w i t h a c o m u l t i p i i c a t i o n . S i n c e c a t X < 1 i f a n d o n l y i f X i s a c o - H s p a c e , t h e n as a c o r o l l a r y o f T h e o r e m 2 . 8 , we h a v e D E F I N I T I O N 2 . 1 8 : A c o m u l t i p i i c a t i o n a on X i s h o m o t o p y a s s o c i a t i v e , i f t h e s q u a r e THEOREM 2 . 1 7 : I f X i s a c o - H s p a c e , t h e n s o i s B X . a X X V X a X V X x v x v x i s h o m o t o p y c o m m u t a t i v e . D E F I N I T I O N 2 . 1 9 : A map T : X — ^ X ( o f t h e c o - H s p a c e X) i s c a l l e d h o m o t o p y i n v e r s e , i f e a c h c o m p o s i t e X —2-— X v X ^ X a n d X ° >• X v X i s h o m o t o p i c t o * : X — X . D E F I N I T I O N 2 . 2 0 : A c o - g r o u p i s a s y s t e m ( X , a , x ) c o n s i s t i n g o f a c o - H s p a c e X w i t h h o m o t o p y a s s o c i a t i v e c o -m u l t i p l i c a t i o n a a n d h o m o t o p y i n v e r s e T . The f o l l o w i n g p r o p e r t y o f s m a s h f u n c t o r w i l l be u s e d f r e q u e n t l y , Z ( X v Y ) = Z X v ZY [ 1 0 ] . T h i s p r o p e r t y e n a b l e s us t o i d e n t i f y B ( n X ) w i t h n ( B X ) v i a a n a t u r a l h o m e o m o r p h i s m k n . L e t ( X , a , x ) be a c o - g r o u p . D e f i n e a = k 2 o Ba BX Ba B ( X V X) B X V BX a n d x = Bx BX BX we s h a l l show t h e f o l l o w i n g p r o p e r t i e s ( 1 ) a n d ( 2 ) . and ( 1 ) T i s a h o m o t o p y i n v e r s e o f In f a c t , by h y p o t h e s i s , we h a v e V o C l V - O o c r ^ * V O ( T V 1 ) O O ~ * A T h e n f r o m t h e c o m m u t a t i v e d i a g r a m - 1 R y V x  B X ° — = ~ B X V BX — BX V BX B ( X y x) B ( l V T ) * - — s » B ( X V X) we h a v e V o 0 B X V x ) o a = BV o B ( 1 X V x ) V o ( 1 x V T ) =: B * = * s i m i l a r l y , V o (x v l B X ) o a (2) a i s h o m o t o p y a s s o c i a t i v e . By h y p o t h e s i s ( l v V c r ) o o - ( o V l ) o A A T h e n f r o m t h e c o m m u t a t i v e d i a g r a m a BX BX ^ B X v BX s - B X V B X V B X B ( l V a ) B ( X V X) * ^ - B ( X V X V X ) a n d a s i m i l a r o n e , we h a v e ( 1 B X V o ) o a k 3 o B ( l v a ) o Ba A = k 3 o B - k 3 o B ( l x V a ) o a ( a V 1 x ) o a C o n s e q u e n t l y we h a v e = k 3 o B ( a V 1 ) o Ba A = ( a " V 1 R X ) o a 35 THEOREM 2 . 2 1 : L e t B , X be p o i n t e d t o p o l o g i c a l s p a c e s . I f ( X , a , x ) i s a c o - g r o u p , t h e n s o i s ( B X , a", T) . A g a i n we h a v e p r e s e r v a t i o n o f s t r u c t u r e ; s e e R e m a r k 2 . 1 4 . D E F I N I T I O N 2 . 2 2 : L e t ( X , a, x ) be a c o - g r o u p . The b a s i c c o - c o m m u t a t o r map i>2 i s t h e c o m p o s i t i o n 2 1 v 2 x X x v x ^ > 2 X v 2 X —*- * 2X v 2 X 2 X T h e c o - c o m m u t a t o r map tyx o f w e i g h t 1 i s t h e i d e n t i t y map o f X , t h e c o - c o m m u t a t o r map ^ n + 1 ° f w e i g h t n+1 i s t h e compos i t i o n x - i * x v x ^ £ J ! D . x v n x - n + 1 x . D E F I N I T I O N 2 . 2 3 : The n i l p o t e n c y n i l ( X , a , x ) o f a c o - g r o u p i s t h e l e a s t i n t e g e r n £ 0 f o r w h i c h t h e map < P n + 1 1 S n u l 1 h o m o t o p i c r e l . b a s e p o i n t s . I f no s u c h i n t e g e r n e x i s t s , we p u t n i l ( X , a , x ) = °° . S u p p o s e , as i n T h e o r e m 2 . 2 1 , t h a t ( X , a , x) a c o - g r o u p . L e t 1j72 be t h e b a s i c c o - c o m m u t a t o r map o f ( B X , a , x) . I t i s e a s y t o s e e t h a t tyz = k 2 o BiJ; 2 S u p p o s e now we h a v e ^ n = k n 0 B ^ n i n t n e d i a g r a m 37 T h e n t h e d i a g r a m Bit? B( 1 V i> ) BX B(XV X) x Vi B ( X V n X ) t l n y \ / ( B * ) B X V B X — ^ 2 - B X V B ( n X ) i d t f BXVBX — — — B X V n ( B X ) i s c o m m u t a t i v e a n d we h a v e Vn = ] B X V *n 0 * 1 B X V k n o k o 6 1 V i> x v n o Bty = k n + l ° B * n + 1 T h u s we h a v e s h o w n i n d u c t i v e l y t h a t i> = k o B f o r J n n n a r b i t r a r y f i n i t e n ( n = 1 i s t r i v i a l ) . 38 THEOREM 2 . 2 4 : L e t B , X be p o i n t e d t o p o l o g i c a l s p a c e s , a n d l e t ( X , a , x ) be a c o - g r o u p . T h e n n i l ( B X , a , x ) < n i l ( X , a , x ) . Proof: n i l ( X , a , x ) = n =^ > ^ n + 1 = * H e n c e \p = k ^. o Bip _,, - k o B* = * r n + l n+1 n+1 n+1 I t i s w e l l - k n o w n t h a t , f o r a p o i n t e d s p a c e X , t h e r e d u c e d s u s p e n s i o n EX i s a c o - g r o u p . I t s c o m u l t i p i i c a t i o n a : EX EX V EX i s d e f i n e d by t h e f o r m u l a o ( [ x , t ] ) = -( [ x , 2 t ] , * ) (* , [ x , 2 t - l ] ) 0 < t < \ \ < t < 1 a n d i t s h o m o t o p y i n v e r s e i s d e f i n e d by T ( [ X , t ] ) = [ x , 1 - t ] 0 < t < 1 39 F o r a r b i t r a r y p o i n t e d s p a c e X , we now d e f i n e i t s c o n i l p o t e n c y by c o n i l X = n i l ( E X , a , x ) , t h e n we h a v e , as an i m m e d i a t e c o n s e q u e n c e o f T h e o r e m 2 . 2 4 c o n i 1 BX < com* 1 X . 40 C h a p t e r 3 THE FUNCTOR A x ~ AND SOME G E N E R A L I Z A T I O N S A L e t A a n d X be two p o i n t e d t o p o l o g i c a l s p a c e s , c o n s i d e r A + = A ( d i s j o i n t u n i o n w i t h {+} as b a s e p o i n t ) . T h e n we h a v e A x X / A = A + x X / { + } x X V A + x * = A + X . I n t h e l a s t c h a p t e r we h a v e s h o w n t h a t t h e s m a s h f u n c t o r B -d o e s n o t i n c r e a s e weak c a t e g o r y , L u s t e r n i k - S c h n i r e l m a n n c a t e g o r y , s t r o n g c a t e g o r y a n d c o - n i 1 p o t e n c y . S i n c e Ax - / + ' A = A - , i t f o l l o w s t h a t t h e s e a r e a l s o n o t i n c r e a s e d Ax - / u n d e r t h e f u n c t o r ' A . H o w e v e r f o r s u c h a f u n c t o r we c a n i m p r o v e t h e r e s u l t o f C o r o l l a r y 2 . 1 2 as f o l l o w s . THEOREM 3 . 1 : L e t A a n d X be c o n n e c t e d p o i n t e d AxX / C W - c o m p l e x e s w i t h A l o c a l l y c o m p a c t , t h e n c a t X = c a t / A . 41 Proof: We n e e d t o show t h a t c a t X < k ^=%> c a t A X X / A < k . The a n d 7T s* " p a r t has b e e n s h o w n i n c o r o l l a r y 2 . 1 2 . To show l e t i : X — ^ be i n d u c e d by i n c l u s i o n Y —s*- X be i n d u c e d by p r o j e c t i o n r e s p e c t i v e l y , w i t h TI o i d i a g r a m = 1 By lemma 2 . 9 we h a v e a c o m m u t a t i v e x , k TT "x , k x , k 'y.k • y , k x , k 7T S u p p o s e t h e r e e x i s t s a s e c t i o n S^ : Y E y , k w i t h P i o S 1 , t h e n c l e a r l y S „ y J x T r k o S y o i i s a s e c t i o n X — * - E . . The t h e o r e m i s now e s t a b l i s h e d by an a p p e a l t o t h e o r e m 2 . 1 1 . Si 42 COROLLARY 3 . 2 : U n d e r t h e same a s s u m p t i o n on A , X as i n t h e o r e m - 3 . 1 , X i s a c o - H s p a c e i f a n d o n l y i f A * X / A 1, . B e f o r e p r o v i n g t h e n e x t t h e o r e m , we n e e d some f a c t s . ( a ) L e t X be a c o - H s p a c e w i t h c o m u l t i p i i c a t i o n o , a n d l e t f i , f 2 : X — — Y be two m a p s , t h e n we c a n f o r m t h e i r sum f i + f 2 : X — — Y by f i + f 2 = <f 1 . f z> cr X 0 > X V X < f 1 ' f z £ Y . Upon a p p l y i n g a h o m o l o g y f u n c t o r H t o t h i s m a p , we h a v e ( f i + f 2 ) * = f i * + f 2 * : H ( X ) H ( Y ) . I n f a c t , s i n c e X i s a c o - H s p a c e , we h a v e X — ^ X V X P i = < 1 » ° ^ X , X — ^ X V X P ^ 0 ' 1 ^ x w i t h p 1 a = 1 , p 2 a = l y T h e n 1 = p . a * : H ( X ) — ^ H ( X V X ) = H ( X ) © H ( X ) - ^ ~ H ( X ) w i t h j = 1 o r 2 . H e n c e f o r a n y x e H ( X ) , we h a v e a* ( x ) = ( x , x ) . On t h e o t h e r h a n d , f r o m t h e d i a g r a m we h a v e f i * ( x ) = < f : , f 2 > * i i * ( x ) = < f i , f 2 > * ( x,0) f o r x e H ( X ) . S i m i l a r l y f z * ( x ) = < f i , f 2 > * i 2 * ( x ) = < f i , f 2 > * ( 0 , x ) . 44 C o n s e q u e n t l y ( f i + f 2 ) * ( x ) = <f i , f 2 > * a* ( x ) = < f i , f 2 > * ( x , x ) = < f i , f 2 > * ( x , 0 ) + < f ! , f 2 > * ( 0 , x ) = f i * ( x ) + f 2 * ( x ) . ( b ) L e t A a n d X be p o i n t e d C W - c o m p l e x e s w i t h A 0 - c o n n e c t e d a n d X 1 - c o n n e c t e d . C o n s i d e r t h e s e q u e n c e o f m a p s , A — 1 — A x X ^ C. , w h e r e i i s t h e i n c l u s i o n a n d C^ i s t h e r e d u c e d m a p p i n g c o n e , h e n c e By Van K a m p e n ' s t h e o r e m , we h a v e 45 TT l (AxXy ) ' A { i * * i ( A ) } J TTi ( A ) x e { ^ ( A ) } N = e [6] C l e a r l y i s p a t h w i s e c o n n e c t e d , h e n c e i t i s 1 - c o n n e c t e d D E F I N I T I O N 3.3: I f X a n d X a r e c o - H s p a c e s w i t h c o m u l t i p i i c a t i o n a a n d a' r e s p e c t i v e l y , a map f : X — s » x ' i s c a l l e d a h o m o m o r p h i s m o r p r i m i t i v e i f t h e s q u a r e X V X f V f x' g > x ' v x ' i s h o m o t o p y c o m m u t a t i v e We a r e now r e a d y t o p r o v e 46 THEOREM 3 . 4 : L e t A a n d X be p o i n t e d C W - c o m p l e x e s w i t h A O - c o n n e c t e d a n d X 1 - c o n n e c t e d . S u p p o s e f u r t h e r t h a t X i s a c o - H s p a c e . T h e n t h e r e i s a h o m o t o p y e q u i v a l e n c e g : A x X / A * X V ( A X ) , a n d g i s n a t u r a l w . r . t . h o m o m o r p h i s m f : X Proof: C o n s i d e r t h e p r o j e c t i o n s Ax <AX S i n c e i s a c o - H s p a c e , t h e r e i s a c o m u l t i p i i c a t i o n A x X / A ^ A x X / A v A x X / A a i n d u c e s g = p + g = ( p v q ) a : A x X / A ^ X V ( A X ) . T h i s i n d u c e s a h o m o l o g y h o m o m o r p h i s m , g * = (p + q ) * : H 'AxX, ^ H X V (AX) 47 S i n c e (p + q ) * = P * + q * » i t f o l l o w s f r o m t h e K l i n n e t h f o r m u l a t h a t g * i s an i s o m o r p h i s m . Now A x X / A i s 1 - c o n n e c t e d , X V ( A X ) i s t r i v i a l , b e i n g A X i s 2 - c o n n e c t e d [ 1 0 ] , a n d TTI t h e f r e e p r o d u c t o f TT i (X) = e a n d T T I ( A X ) = e . T h e r e f o r e by a t h e o r e m o f J . H . C . W h i t e h e a d , g i s a h o m o t o p y e q u i v a l e n c e The n a t u r a l i t y o f g w . r . t . homomorph i sms f : X — » - x' i s t r i v i a l . B I n t h e r e s t o f t h i s c h a p t e r we d i s c u s s some a b s t r a c t g e n e r a l i z a t i o n s o f t h e s m a s h f u n c t o r B - . L e t F be a n y f u n c t o r on * w h i c h has t h e f o l l o w i n g p r o p e r t i e s . 1 ) F p r e s e r v e s n u l l o b j e c t s , n u l l maps a n d h o m o t o p i e s . 2 ) L e t T n ( A ) = { ( a i , a 2 , • . • , a n ) e A i X • • • x A n | a t l e a s t o n e a^ = *} be t h e f a t wedge a n d l e t u = { F p i , • • • , F p n } : F ( A x x • • • x A n ) ^ F A i X • • • x F A n . The map u> i s c h a r a c t e r i z e d by t h e c o m m u t a t i v i t y o f t h e d i a g r a m s 48 We r e q u i r e t h a t to i n d u c e i|> : F ( T n A ) — ~ T n ( F A ) , so t h a t t h e d i a g r a m > ( A ) ] -i> T n ( F A ) F(AiX x A n ) . F A i X • • • x F A , i s c o m m u t a t i v e , w h e r e j a n d j a r e i n c l u s i o n s . I f A.j = X f o r a l l i , t h e n by p r o p e r t y 2 ) , a n d u s i n g t h e same a r g u m e n t as i n t h e o r e m 2 . 8 , we h a v e 49 THEOREM 3 . 5 I f X e C 7 ! , t h e n c a t FX < c a t X N e x t , a s s u m e t h a t F s a t i s f i e s 1) a n d 2 ) , a n d t h a t i n a d d i t i o n F p r e s e r v e s c o k e r n e l s . T h e n we h a v e t h e f o l l o w -i n g c o m m u t a t i v e d i a g r a m : T k + l ( x ) k+l n x i x ( k + l ) ^ T K + 1 ( F X ) CO k+l n F X V 3 * ( F X ) q (k+l) w h e r e q a n d q~ a r e t h e c a n o n i c a l m a p s . The e x i s t e n c e o f <j> i s due t o t h e f a c t F ^ i s t h e c o k e r n e l o f F ^ , a n d q~ J = * . U n d e r t h e s e c i r c u m s t a n c e s we h a v e THEOREM 3 . 6 : I f X e J * , t h e n w c a t FX < w c a t X The p r o o f i s a n a l o g o u s t o t h e p r o o f o f t h e o r e m 2 . 3 . | L e t G be a f u n c t o r on C7* , s u c h t h a t 1 . G p r e s e r v e s n u l l o b j e c t s , n u l l maps a n d h o m o t o p i e s ; 2 . G p r e s e r v e s f i n i t e c o p r o d u c t s , i . e . G i s a d d i t i v e T h e n as i n t h e o r e m 2 . 2 4 , we h a v e THEOREM 3 . 7 : I f X i s a c o - g r o u p i n , t h e n c o n i l GX < c o n i l X . C h a p t e r 4 C O U N T E R E X A M P L E S I n t h i s c h a p t e r we p r e s e n t some e x a m p l e s t o show t h a t some o f o u r p r e v i o u s r e s u l t s a r e no l o n g e r t r u e u n d e r m o r e g e n e r a l h y p o t h e s e s , a n d t h a t some r e s u l t s a r e n o t d u a l i z a b l e . F i r s t we n o t e t h a t t h e s p a c e A X X / A a n d X V A X (A , X a r e b o t h i n U J * ) a l w a y s h a v e t h e same a d d i t i v e h o m o l o g y s t r u c t u r e . I n f a c t , c o n s i d e r t h e s h o r t s e q u e n c e o f m a p s : A — ^ A x X A x X / A , w i t h i t h e i n c l u s i o n m a p , p t h e p r o j e c t i o n , a n d p i = 1^ The r e d u c e d h o m o l o g y s e q u e n c e s p l i t s a c c o r d i n g l y , ~ ~ a ~ H ( A ) ^ H n ( A x X ) — ^ H n ( A x X , A) -2- H n _ ] ( A ) P * 52 g i v i n g H n ( A x X) = H n ( A ) © H n ( A x X , A) Now Ax X A = H A x X , A = H Ax X , A f o r a l l n > 0 [ 1 0 ] , s o t h a t Ax X . 1 ^ H (Ax X ) / I 7 A J " " / H n ( A ) f o r a l l n > 0 On t h e o t h e r h a n d , f r o m t h e f o r m u l a we h a v e H n ( A x X) = H n ( A ) e H n ( X ) © H n ( A X ) , H „ ( X V A X ) = H ( X ) © H n ( A X ) = H n ( A x X j / . n f o r a l l n > 0 53 H e n c e t h e two s p a c e s i n q u e s t i o n h a v e t h e same h o m o l o g y g r o u p s . C o n s e q u e n t l y t h e i r a d d i t i v e c o h o m o l o g y g r o u p s a r e a l s o t h e s a m e . N e v e r t h e l e s s , i f t h e c o n d i t i o n " X i s a c o -H s p a c e " i n t h e o r e m 3 . 4 i s w e a k e n e d t o " X i s an a r b i t r a r y 1 - c o n n e c t e d s p a c e , " t h e n A x X A a n d X v AX a r e n o t n e c e s s a r i l y h o m o t o p y e q u i v a l e n t E X A M P L E 4 . 1 : T a k e A = S 1 a n d X = C P 2 . T h e i r s i n g u l a r c o h o m o l o g y r i n g s ( w i t h Z as c o e f f i c i e n t r i n g ) a r e r e s p e c t i v e l y as f o l l o w s : F o r S 1 , we h a v e f H ^ S 1 ) = Z H 1 ( S 1 ) = Z H n ( S 1 ) = 0 n > 1 w i t h e ° e H ^ S 1 ) e 1 e H ^ S 1 ) as g e n e r a t o r s a n d a n d e ° u e ° = e ° , e ^ e 1 = e ^ e 0 = e 1 , e 1 w e 1 = 0 . 54 F o r C P 2 , we h a v e t h a t H * ( C P 2 ) i s t h e r i n g o f p o l y n o m i a l s t r u n c a t e d a t d e g r e e 3, w i t h y e H 2 ( C P 2 ) as a p o l y n o m i a l g e n e r a t o r . L e t us now c o m p a r e t h e m u l t i p l i c a t i v e c o h o m o l o g y s t r u c t u r e o f t h e s p a c e S 1 x C P 2 / Y i — / c i a n d Y 2 = C P 2 V ( S ^ P 2 ) . U s i n g t h e K u n n e t h f o r m u l a a n d n o t i n g t h a t T o r [ Z , 0 ] = 0 = T o r [ Z , Z ] we h a v e H n (Y i ) = © H ^ S 1 ) ® H n _ i ( C P 2 ) / H n ( S 1 ) , i=0 h e n c e H ° ( Y i H 1 ( Y i H 2 ( Y ! H 3 (Y i H * ( Y i H 5 ( Y i Z 0 H ° ( S 1 ) H 1 ( S 1 ) Z H M S 1 ) H 2 ( C P 2 ) - Z H 2 ( C P 2 ) = Z H " ( C P 2 ) - Z The g e n e r a t o r s o f H 2 ( Y i ) , H 3 ( Y i ) a n d H 5 ( Y x ) a r e e ° ® Y > e 1 ® y a n d e 1 ® y 2 r e s p e c t i v e l y , w i t h ( e ° ® y)^{e1 ® y ) = ( e ^ e 1 ) ® ( Y ^ Y ) = e 1 ® y : On t h e o t h e r h a n d , f o r Y 2 , we h a v e a n d ^ ( S ' C P 2 ) = " " ^ X C P 2 > H ^ S 1 ) 8 H N ( C P 2 ) C P 2 v ( S ^ P 2 ) H n ( C P 2 ) © H n ( S 1 C P 2 ) ( D i r e c t sum o f r i n g s ) A g a i n by c o m p u t a t i o n , we g e t H ° ( Y 2 ) H 1 ( Y 2 ) H 2 ( Y 2 ) H 3 ( Y 2 ) H " , ( Y 2 ) H 5 ( Y 2 ) Z 0 H ( C P 2 ) © 0 = Z 0 © H M S ^ P 2 ) = 0 z 0 © H 5(S'CP 2) = 0 H ^ S 1 ) ® H 2 ( C P 2 ) = Z H 1 ( S 1 ) ® H 4 ( C P 2 ) = Z 56 B u t i n t h i s c a s e , t h e g e n e r a t o r s o f H 2 ( Y 2 ) , H 3 ( Y 2 ) a n d H 5 ( Y 2 ) a r e y e H 2 ( C P 2 ) , e 1 0 y e H ^ S ^ P 2 ) a n d e 1 ® y 2 £ H 5 ( S 1 C P 2 ) r e s p e c t i v e l y w i t h y ^ i e 1 ® y ) = 0 + e 1 ® Y 2 • So Y i a n d Y 2 h a v e d i f f e r e n t c o h o m o l o g y r i n g s , t h u s t h e y a r e n o t h o m o t o p y e q u i v a l e n t . N e x t , i n c o r o l l a r y 3 . 2 , a r i s e s as t h e c o f i b r e o f t h e c o f i b r a t i o n A — ^ — 3 - A x X . 3 . 2 c l a i m s t h a t i f X i s a c o - H s p a c e , t h e n so i s t h e c o f i b r e o f i . Now l e t us t u r n t o t h e d u a l q u e s t i o n , i . e . c o n s i d e r t h e f i b r a -t i o n A V X —P—-s* A w i t h f i b r e F . I s F an H - s p a c e when X i s o n e ? The a n s w e r i s , i n g e n e r a l , n e g a t i v e . T h i s c a n be s e e n i n E X A M P L E 4 . 2 : N o t i c e t h a t F = ( E A x *)\J{MK x X) C EA x X w i t h EA = { £ e A 1 I £(!)=*} w h i c h i s c o n t r a c t ! " b l e , so t h a t / QA 57 T a k e X = S 3 w h i c h i s an H - s p a c e a n d A = K ( Z , 2) , so t h a t QA = S 1 , a c c o r d i n g t o t h e o r e m 3.4. B u t s 3 v s^ i s n o t an H - s p a c e C o n s i d e r t h e c o f i b r a t i o n s A — — A x X — K a n d A — 2 — A x EX K ' w i t h c o f i b r e s K a n d K r e s p e c t i v e l y . We t h e n h a v e K ' = A X Z X / A _ Z (A x X . ) 1 A Z K The d u a l o f t h e a b o v e i m p l i c a t i o n i s P A ~ P A F =- A v x — * w A a n d F - A v QX — ^ - A 58 w i t h f i b r e s F a n d F r e s p e c t i v e l y , we h a v e F fiF . B u t t h i s i s f a l s e ( e v e n i f X i s an H - s p a c e ! ) as we s e e i n E X A M P L E 4 . 3 : T a k i n g A = K(Z , 2) = X , f J A x M / ^ s ' x s 1 / , 2 a n d F ^ ftF b e c a u s e QF i s an H - s p a c e . F i n a l l y we p r o v i d e an e x a m p l e t o show t h a t i n g e n e r a l , c a t BX £ M i n ( c a t B , c a t X) . E X A M P L E 4 . 4 : T a k e B = K ; (Z , 2 ) X = K' (Z , 2 ) , t h e M o o r e p o l y h e d r a w i t h g c d (m , n) = 1 . S i n c e B - ZK' (Z , 1 ) , X = ZKF (Z , 1) , t h e y h a v e t h e h o m o t o p y t y p e o f s u s p e n s i o n s ( a n d i n p a r t i c u l a r , t h e y a r e 1 - c o n n e c t e d ) . H e n c e c a t B = C a t B = 1 a n d c a t X = C a t X = 1 59 9 Now c o n s i d e r t h e s h o r t s e q u e n c e B V X >—-—* B x X » B X we h a v e H k ( B V X ) = H k ( B ) 0 H k ( X ) = 0 k + 2 , 0 H 0 ( B V X) = Z H 2 ( B V x) = Z m © Z n . On t h e o t h e r h a n d , i t i s w e l l - k n o w n t h a t a n d T o r ( Z m , 1) = Z , . = 0 m n ' (m , n ) Z ® Z = 1, x = 0 m n (m , n ) By t h e K u n n e t h f o r m u l a , we g e t H k ( B x x ) = 0 k + 2 , 0 H 0 ( B x x) = Z a n d H 2 ( B x X) = Z m ® Z © Z ® Z n = Z m © Z n . 60 T h e r e f o r e , i i s a h o m o t o p y e q u i v a l e n c e . H e n c e BX - * , w h i c h i m p l i e s c a t BX = 0 1 M i n ( c a t B , c a t X ) . REMARK 4 . 5 : I n g e n e r a l , f o r a p o i n t e d s p a c e X , we h a v e ( 1 ) w c a t X < c a t X < C a t X , a n d ( 2 ) c o n i l ( X ) < w c a t ( X ) [ 1 1 ] ( w h e r e X i s a H a u s d o r f f s p a c e w i t h n o n d e g e n e r a t e b a s e p o i n t ) . F o r c o m p l e t e n e s s , we a l s o c o l l e c t two e x a m p l e s t o show t h e s t r i c t i n e q u a l i t i e s i n ( 1 ) . E X A M P L E 4 . 6 : L e t X be t h e s p a c e o b t a i n e d by r e -m o v i n g an o p e n 3 - c e l l f r o m a P o i n c a r e 3 - s p h e r e . S i n c e T T I ( X ) =f e a n d i s n o t f r e e , we h a v e c a t X > 1; ( c a t X = 1 7T iX i s f r e e ) . On t h e o t h e r h a n d , X ^ 2 ^ i s 1 - c o n n e c t e d . S i n c e ^ • ( X ) 0 f o r a l l i , we g e t H, (2) = 0 H e n c e (2) 61 i s c o n t r a c t ! ' b l e , c o n s e q u e n t l y q A * * : X — X x X q X ( 2 ) i . e . w c a t X = 1 . E X A M P L E 4 . 7 : I t i s w e l l known t h a t c a t X < 1 i f f X i s a c o - H s p a c e , a n d C a t X < 1 i f f X h a s t h e h o m o t o p y t y p e o f s u s p e n s i o n . T a k i ng X = s 3 ^ e 2 p + 1 , a w h e r e a i s o f o r d e r p w i t h p an o d d p r i m e , t h e n X i s a c o - H s p a c e , b u t i t i s n o t a s u s p e n s i o n . H e n c e c a t X < 1 , b u t C a t X > 1 [ 1 ] 62 R E F E R E N C E S [ 1 ] I . B e r s t e i n a n d P . J . H i l t o n , C a t e g o r y a n d g e n e r a l i z e d H o p f i n v a r i a n t s , I l l i n o i s J . M a t h . ( 1 9 6 2 ) , p p . 4 3 7 -451 . [ 2 ] I . B e r s t e i n a n d T . G a n e a , H o m o t o p i c a l n i l p o t e n c y , I l l i n o i s J . M a t h . ( 1 9 6 1 ) , p p . 9 9 - 1 3 0 . [ 3 ] T . G a n e a , L u s t e r n i k - S c h n i r e l m a n n c a t e g o r y a n d s t r o n g c a t e g o r y , I l l i n o i s J . M a t h . , V o l . 1 1 , N o . 3 ( 1 9 6 7 ) , p p . 4 1 7 - 4 2 7 . [ 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 , a n d M a t h e m a t i c a l P h y s i c s , N o . 43 ( 1 9 5 3 ) . [ 5 ] P . J . H i l t o n , H o m o t o p y T h e o r y a n d D u a l i t y , G o r d o n a n d B r e a c h ( 1 9 6 5 ) . [ 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 C o u r s e , L e c t u r e N o t e s , C o u r a n t I n s t i t u t e o f M a t h e -m a t i c a l S c i e n c e s ( 1 9 6 9 ) . [ 7 ] D. P u p p e , H o m o t o p i e m e n g e n u n d i h r e i n d u z i e r t e n A b b i l d u n g e n , I . M a t h e m a t i s c h e Z e i t s c h n i f t , B d . 6 9 , p p . 2 9 9 - 3 4 4 ( 1 9 5 8 ) . [ 8 ] E . H . S p a n i e r , A l g e b r a i c T o p o l o g y , M c G r a w - H i l l ( 1 9 6 6 ) . [ 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 , H o m o l o g y a n d c o h o m o l o g y , W . A . B e n j a m i n I n c . ( 1 9 7 0 ) . [ 1 0 ] G . W . W h i t e h e a d , G e n e r a l i z e d H o m o l o g y T h e o r i e s , T r a n s . A m e r . M a t h . S o c , 1 02 ( 1 9 6 2 ) , p p . 2 2 7 - 2 8 3 . 63 [ 1 1 ] 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 o f l o o p - s p a c e s a n d s u s p e n s i o n s , T o p o l o g y 1 ( 1 9 6 2 ) , p p . 1 3 3 - 1 6 1 . 

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