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 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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