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Some computations of the homology of real grassmannian manifolds Jungkind, Stefan Jörg 1979

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9i< SOME COMPUTATIONS OF THE HOMOLOGY OF REAL GRASSMANNIAN MANIFOLDS by  STEFAN JORG JUNGKIND B . S c , The U n i v e r s i t y o f A l b e r t a , 1977  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics)  We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1979  ©  S t e f a n J o r g J u n g k i n d , 1979  : - 6  In p r e s e n t i n g  this thesis i n partial  f u l f i l m e n t o f the r e q u i r e m e n t s f o r  an advanced degree a t the 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 agree t h a t the 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 and s t u d y .  I f u r t h e r agree 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  copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department 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 understood t h a t c o p y i n g o r p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l written  gain s h a l l  permission.  Department n f  Hg^it^aj-'ics  The U n i v e r s i t y o f B r i t i s h Columbia 207.5 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  B P  7 5 - 5 1 1 E  not be a l l o w e d w i t h o u t my  Abstract When computing t h e homology o f Grassmannian m a n i f o l d s , t h e f i r s t s t e p i s u s u a l l y t o l o o k a t t h e S c h u b e r t c e l l d e c o m p o s i t i o n , and t h e c h a i n complex a s s o c i a t e d w i t h i t . case w i t h  Z^  I n t h e complex case and t h e r e a l u n o r i e n t e d  c o e f f i c i e n t s the a d d i t i v e s t r u c t u r e i s obtained immediately  ( i . e . , g e n e r a t e d by t h e homology c l a s s e s r e p r e s e n t e d by t h e S c h u b e r t c e l l s ) because t h e boundary map i s t r i v i a l . Z^  I n t h e r e a l u n o r i e n t e d case ( w i t h  c o e f f i c i e n t s ) and t h e r e a l o r i e n t e d c a s e , f i n d i n g t h e a d d i t i v e  i s more c o m p l i c a t e d s i n c e t h e boundary map i s n o n t r i v i a l .  structure  In t h i s paper,  t h i s boundary map i s computed by c e l l o r i e n t a t i o n c o m p a r i s o n s , u s i n g graph c o o r d i n a t e s where t h e c e l l s a r e l i n e a r , t o s i m p l i f y t h e c o m p a r i s o n s .  The  i n t e g r a l homology groups f o r some l o w d i m e n s i o n a l o r i e n t e d and u n o r i e n t e d Grassmannians a r e d e t e r m i n e d d i r e c t l y from t h e c h a i n complex ( w i t h t h e boundary map a s computed). The i n t e g r a l cohomology r i n g s t r u c t u r e f o r complex Grassmannians has been c o m p l e t e l y d e t e r m i n e d m a i n l y u s i n g S c h u b e r t c e l l (what i s known a s S c h u b e r t C a l c u l u s ) . .  I n t h i s p a p e r , a method  Schubert c e l l i n t e r s e c t i o n s t o d e s c r i b e t h e of t h e r e a l Grassmannians i s s k e t c h e d . for  t h e complex Grassmannians ( w i t h  Z^  cohomology r i n g  using structure  The r e s u l t s a r e i d e n t i c a l t o t h o s e c o e f f i c i e n t s ) , but the n o t a t i o n  used f o r t h e cohomology g e n e r a t o r s i s n o t t h e u s u a l one. the  intersections  I t indicates that  p r o d u c t s a r e t o a c e r t a i n degree independent o f t h e Grassmannian.  iii  TABLE OF CONTENTS Page Abstract Table o f Contents  i  L i s t o f Tables  i  i  i  i iv  L i s t o f Figures  v  Acknowledgement  v i  Introduction  1  PART I - DEFINITIONS AND NOTATION  3  Grassmannian M a n i f o l d s and Mappings between Them  3  Schubert C e l l s and Schubert V a r i e t i e s  6  . . .  Graph C o o r d i n a t e s and Chain Complexes f o r t h e Grassmannians  . .  PART I I - ADDITIVE HOMOLOGY STRUCTURE  10 15  G e n e r a l Theory f o r C e l l Complexes  15  D e t e r m i n i n g t h e I n c i d e n c e Number f o r t h e Boundary Map f o r C(G, ) l k, n Some Low D i m e n s i o n a l Examples  17 25  T a b l e s o f Homology Groups o f t h e Grassmannians  33  PART I I I - HOMOLOGY AND COHOMOLOGY PRODUCTS  37  The G e n e r a l I n t e r s e c t i o n Theory To Be Used Simple I n t e r s e c t i o n s i n G, and t h e P o i n c a r e D u a l i t y Map K5n C o m p l i c a t e d I n t e r s e c t i o n s and t h e G e n e r a l Formula References  37 . .  38 48 57  iv  L i s t o f Tables I.  The Boundary Map i n C(G) . .  27  IV  II. III. IV.  Homology Groups f o r G .  34  Low D i m e n s i o n a l Homology Groups f o r G  35  Homology f o r t h e u n o r i e n t e d Grassmannians  36  V  L i s t of Figures 1.  ( R e f e r r i n g t o Example 2.4.)  17  2.  ( R e f e r r i n g t o the p r o o f o f P r o p o s i t i o n 2.8.)  21  vi  Acknowledgement I wish to express thanks t o my supervisor Professor Mark Goresky f o r h i s help with the m a t e r i a l presented, and a l s o t o Professors Kee Y. Lam and Jim C a r r e l l f o r t h e i r a d d i t i o n a l a i d i n research. This paper was w r i t t e n while on a scholarship from the N a t i o n a l Research Council of Canada.  1  Introduction A l o t i s known about t h e homology  o f t h e Grassmannian m a n i f o l d s i n  g e n e r a l ; e.g., from C h a r a c t e r i s t i c C l a s s e s (see [1]) o r u s i n g a l g e b r a i c g e o m e t r i c methods (see [3] and [ 4 ] ) and u s u a l l y t h e Schubert c e l l d e c o m p o s i t i o n i s used.  However, t h e r e do n o t seem t o be r e a d i l y  available  answers t o such q u e s t i o n s a s : a) G i v e n a f i n i t e Grassmannian, what i s t h e  dimensional oriented or unoriented r e a l r - t h homology  group?  b) G i v e n two c o c y c l e s i n such a Grassmannian, what i s t h e i r cup p r o d u c t ? T h i s paper i s concerned w i t h d e v e l o p i n g c o m p u t a t i o n a l methods, u s i n g t h e geometry o f t h e Schubert c e l l d e c o m p o s i t i o n , by which e x p l i c i t answers t o the above can be d e t e r m i n e d . In P a r t I I , ( a ) i s t a c k l e d by c o n s t r u c t i n g two U n i v e r s a l c h a i n complexes a r i s i n g from t h e Schubert c e l l d e c o m p o s i t i o n o f t h e U n i v e r s a l o r i e n t e d ( r e a l ) Grassmannian and t h e U n i v e r s a l u n o r i e n t e d Grassmannian. (The main p o i n t i s t o compute t h e boundary maps.) i n t e g r a l homology groups o f some o f t h e f i n i t e  From t h e s e complexes, t h e  Grassmannians and, i n low  d i m e n s i o n s , f o r t h e i n f i n i t e Gras smannians a r e c a l c u l a t e d .  Theoretically,  i t s h o u l d be p o s s i b l e t o determine a l l t h e homology groups f o r a l l t h e r e a l Grassmannians ( o r i e n t e d and u n o r i e n t e d ) from t h e f o r m u l a s g i v e n f o r t h e boundary maps, but t h e amount o f c a l c u l a t i o n r e q u i r e d i n c r e a s e s r a p i d l y i n the  h i g h e r dimensions (above 6 f o r i n s t a n c e ) .  However, by l o o k i n g a t t h e  l o w e r d i m e n s i o n s , i t may be p o s s i b l e t o d e t e c t p a t t e r n s and make c o n j e c t u r e s which c o u l d be proved by o t h e r means, e.g., c h a r a c t e r i s t i c c l a s s e s .  On t h e  2 o t h e r hand, comparing what i s known about c h a r a c t e r i s t i c c l a s s e s w i t h what i s o b t a i n e d here may y i e l d f u r t h e r  i n f o r m a t i o n about t h e c h a r a c t e r i s t i c  c l a s s e s , e.g., w h i c h S c h u b e r t c e l l s c o r r e s p o n d t o a g i v e n c h a r a c t e r i s t i c class.  Homology f o r a r b i t r a r y Schubert v a r i e t i e s can be d e t e r m i n e d from t h e  c h a i n complexes a l s o , and some examples a r e g i v e n . Question  (b) i s c o m p l e t e l y s o l v e d i n i n t e g r a l homology f o r t h e  complex Grassmannians i n [3] (pages 1072-1073), and t h e  ZQ-cohomology r i n g  f o r t h e i n f i n i t e u n o r i e n t e d Grassmannians i s known ( [ 1 ] page 83 and  [5]  page 52) and some o f t h e f i n i t e u n o r i e n t e d Grassmannians ( [ 5 ] page 5 1 ) . Part I I I i t i s indicated,  using i n t e r s e c t i o n p r o d u c t s , t h a t the formulas i n  [3] f o r t h e complex case a r e v a l i d a l s o i n u n o r i e n t e d case.  The  Z^  Z^  cohomology f o r t h e r e a l  cohomology p r o d u c t s i n t h e u n o r i e n t e d S c h u b e r t  v a r i e t i e s can be d e t e r m i n e d a l s o , u s i n g t h e i n d u c e d map t h e i r embeddings  In  i n cohomology o f  i n Grassmannians.  The i n t e r s e c t i o n methods used a r e a f i r s t s t e p i n f i n d i n g  products  i n i n t e g r a l cohomology o f o r i e n t e d and u n o r i e n t e d Grassmannians, but t h i s i s a much more c o m p l i c a t e d problem ( m a i n l y because o f s i g n s ) and w i l l n o t be l o o k e d a t i n t h i s paper.  3  PART I - DEFINITIONS AND NOTATION  Grassmannian M a n i f o l d s and Mappings Between Them  1.1  Definition:  i ) The r e a l u n o r i e n t e d f i n i t e Grassmannian  G,  i s the  k, n set  of  k+n R ,  k  d i m e n s i o n a l p l a n e s through t h e o r i g i n ( c a l l them  w i t h t o p o l o g y g i v e n as f o l l o w s : Let  V  v e c t o r s i n R^ . +n  i t s topology. (where  A  and  be t h e s e t o f o r d e r e d V, k, n  D e f i n e an e q u i v a l e n c e r e l a t i o n B  t o t h e rows o f G,  k-tuples of l i n e a r l y  i s an open subset o f  are  k x (k + n)  t r a n s f o r m a t i o n o f t h e row space o f A  k-planes) i n  B  (i.e.,  = the quotient of  x  ~  n +  A  on  n  j -t  n u s  V, as K,n  inherits  A ~ B  i f they have t h e same row s p a c e ) .  by  defines the topology f o r G Grassmannian  G. k, n  k+n k-planes  a  onto i t s e l f which maps t h e rows o f  i i ) The r e a l o r i e n t e d f i n i t e  oriented  k)  matrices) i f there i s a l i n e a r  A ~ B V  j^ (  independent  (through t h e o r i g i n ) i n R  i s the set o f  w i t h t o p o l o g y g i v e n as  follows. D e f i n e an e q u i v a l e n c e r e l a t i o n  on  V  a s above except t h a t  the l i n e a r t r a n s f o r m a t i o n must have p o s i t i v e determinant. G = the quotient o f V by *a d e f i n e s t h e t o p o l o g y f o r G K,n  K,n  Kj  n  k+n 1.2  Notation:  k x (k + n)  i ) A representation f o ra  matrix  A  h a v i n g row space  k-plane  P  i n R  isa  P. k+n  i i ) A r e p r e s e n t a t i o n f o r an o r i e n t e d  k-plane  P  in R  above, w i t h t h e a d d i t i o n a l c o n d i t i o n t h a t t h e o r i e n t a t i o n determined o r d e r e d row v e c t o r s o f  A  c o i n c i d e s with the o r i e n t a t i o n o f  P.  i s as by t h e  4  1.3  Remarks:  i ) G, i s always o r i e n t a b l e , b u t k, n [2] o n l y i f k + n i s even).  (from  i i ) There i s a n i n v o l u t i o n  plane  Notation: Q  i s not i n general  G, which t a k e s a n o r i e n t e d k, n t o t h e same p l a n e w i t h o p p o s i t e o r i e n t a t i o n .  P  Call  i s antipodal to  T  T  G, k, n  on  t h e a n t i p o d a l map, and i f Q = T ( P ) say t h a t  P.  ( I n G. = S , T i s t h e u s u a l a n t i p o d a l map.) l ,n i i i ) There i s a double c o v e r i n g \|r : G G, which t a k e s a n K 5 ri  K jn  oriented plane  P  t o t h e same p l a n e  P  with o r i e n t a t i o n ignored  (\|c  identifies antipodal points). Note: 1.4  ( i i ) and ( i i i ) show t h a t  G  i sa  Z  bundle o v e r  G.  Mappings between t h e Grassmannians: The n o t a t i o n used here w i l l be used throughout t h e paper. i ) For  of  i 5 p,  J(P)  R.  p = k + n  •+ G  j : G  G  i n R^  f : R  and  j : G  1 5 i < p.  •+ R  P  for p - q  q  q = k + n'  ,  (n' > n),  by  j  by  j ( P ) = t h e k-plane  by  j  (oriented plane  j(e\) =  ,  1 : R  For p = k + n  G  • • • > kt_ e  a k  n  d  b  y  1  1(P)  P  -+ R ,  and  (  P  )  p > q,  q  =  q = k' + n  t  h  e  k  '~P  l a n e  1 5 i 5 p.  i n d u c e s embeddings  j(P) i n R  P) = t h e p l a n e  , and  j(P) i n  w i t h o r i e n t a t i o n i n d u c e d by t h e o r i e n t a t i o n o f  +n  1 : G, -*• G k,n^ k',n l»  i - t h standard b a s i s vector  P  i i i ) Define  e  w i l l denote t h e  I  i i ) Define For  e.  P.  by  l(e.) = e l q-p+i ( k ' > k ) , 1 i n d u c e s embeddings  i n R ' k  + n  spanned by  5  and 1 : G, +  n  -»• G, ,  spanned by  by  1  ( o r i e n t e d plane  e-^, . . . , ]<'_k e  determined by t h e o r i e n t a t i o n o f orientation of  *  i ( P ) , with o r i e n t a t i o n  <e~^, . . . ,  !_]<•> f o l l o w e d by t h e  1 ( P ) i n d u c e d by t h e o r i e n t a t i o n o f  Note:  G  a n c  P) = t h e k'-plane i n  i f k' > k  and  n' > n  > k,n'  k,n  G  1  G  1  1  t h e n t h e diagrams  T  7~* k , n '  k,n  G  and  I k',n~— * k ' , n G  G  i v ) F o r each  | „— P  :  ]<  G  G n  k, n  n,k  P.  commute.  t h e r e a r e homeomorphisms  t a k i n g a plane  P  t o i t s o r t h o g o n a l complement  . k+n in R D  and \,^  -+ G .^  G  n  complement  t a k i n g an o r i e n t e d p l a n e  n  t o i t s orthogonal  P"* o r i e n t e d so t h a t t h e p r o d u c t o r i e n t a t i o n on 1  c o i n c i d e s w i t h t h e s t a n d a r d o r i e n t a t i o n on  1.5  P  Definition:  R^ , +n  i ) The i n f i n i t e u n o r i e n t e d Grassmannian  l i m i t ( v i a t h e embeddings  j : G^ ^ -> n  > n  i)  as  n -> °° o f  i i ) The i n f i n i t e o r i e n t e d Grassmannian ( v i a t h e embeddings Note:  P x p"*~  j ) as  n -+ °° o f  G^  n  G^  i s the union G^  n  .  i s the union  limit  .  These l i m i t s e x i s t s i n c e f o r n S  5 iij ,  t h e diagrams  6  k,n j G  >  V  ,k,n v  S  s  s  s  ^  k,n —*  a  G  ±  k,n  n  commute.  d  3  2  k,  ^  n i  ^k,n  9  By t h e n o t e f o l l o w i n g 1.4 ( i i i ) above, t h e embeddings  1  and  1  induce embeddings  1  :  G  k ~" k ' G  a n d  1  :  G  v ~* k ' G  fo r  k 5 k'.  I t i s easy t o see t h a t here a l s o t h e d i grams a  and  1  commute f o r k 5 k^ < k  2  V* i< G  Thus t h e f o l l o w i n g d e f i n i t i o n s a r e v a l i d : i i i ) The u n i v e r s a l u n o r i e n t e d Grassmannian as  k -+ °° o f  G  G  i s the union l i m i t \  .  k  i v ) The u n i v e r s a l o r i e n t e d Grassmannian k ->• °° o f  G  G  i s t h e u n i o n l i m i t as  .  k  Schubert C e l l s and Schubert V a r i e t i e s  1.6  Definition:  i ) A Schubert symbol  (o-j, . . . , CT ) such t h a t  I Cr|  =  CT  1  +  .  .  . +  0"  k  1  e  Q  in G  k-tuple of integers k  The " d i m e n s i o n " o f CT  .  i i ) G i v e n a Schubert symbol Schubert " c e l l "  isa  0 5 CT 5 . . . 5 o- .  k  IS  CT  k > n  cr such t h a t  t o be t h e s e t o f  a  k  5 n,  k-planes  s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s ( c a l l e d t h e Schubert  define the  P in c o n d i t1 0 n s  ,k+n  7 a s s o c i a t e d w i t h CT):  and  dimension o f  P fl 3 ( R ° ^ ) = i  dimension o f  P fl 3(R ^"  +1  CT  Notation: lies i n e Remark:  F  = i - 1  +1  i s t h e k-plane  G  f o r i = 1, . . . , k. <e  CT +  ^, . . . , e  which  CT  .  CT  The v a l i d i t y o f t h e terms " d i m e n s i o n s " and " c e l l " above w i l l be  shown below. 1.7  Theorem:  Let k > 0  and  n > 0  i ) F o r any Schubert symbol e  Q  c  be g i v e n . cr = (cr^, . . . , cr^ 5 n ) ,  i s an open c e l l o f d i m e n s i o n  n  the set  |cr| .  i i ) The c o l l e c t i o n o f a l l such  e  gives  G  a c e l l complex  v  structure. pf:  see [1] S e c t i o n 6. ( i ) i s proved i n 1.19.  1.8  Proposition:  For e  CT  a Schubert c e l l i n G^  o f a n t i p o d a l open c e l l s i n G^ pf:  ,  f  t o A.  f  other h a l f 1.9  Let e  with orientation (T(e  Corollary:  + C T  CT  .  (A) i s a p a i r o f d i s j o i n t s e t s each homeomorphic  +  CT  e  A c Y is  The p r o p o s i t i o n t h e n f o l l o w s from t h e f a c t t h a t  Notation: P  i sa pair  \Jr t o  i s a double c o v e r i n g and  an open c e l l and t h u s c o n t r a c t i b l e , and t h a t  plane  Q  each homeomorphic under  I n g e n e r a l , i f f : X -> Y  c o n t r a c t i b l e , then under  n  , ^~ (e )  n  CT  e  CT  is  \|/ i d e n t i f i e s a n t i p o d a l p o i n t s ,  -1 denote t h e h a l f o f \|r ( e ) c o n t a i n i n g t h e CT  <^<j^+±> < • < » ^a^+k-*  a n c  ^  e  o ~ denote t h e  )).  Let k > 0  and  n > 0  The c o l l e c t i o n o f open c e l l s  be g i v e n . e  + C T  and  e  CT  where CT r u n s o v e r  a l l S c h u b e r t symbols o f t h e form  (cr-p • • • ,  5 n)  gives  n  a cell  complex s t r u c t u r e . pf: \|r  By 1.8, t h e c e l l s t r u c t u r e f o r G^  to a c e l l structure f o r  G^  g i v e n i n 1.7 p u l l s back v i a  n  made up o f t h e c e l l s  n  e  (J  , e ~  for a l l  and  G  , the  CT  a p p r o p r i a t e Schubert symbols.  1.10  Claim:  W i t h t h e above c e l l s t r u c t u r e s f o r G^  maps _ L , j , 1, JL, j , 1, T  _ l i e  C  T  = e ,  )  j(e ) = e C T  ^  are c e l l u l a r .  where  o^' = t h e number o f  CT  c o n s i d e r e d as a c e l l i n G^ i where  C T  + C T  ) = e  + C T  ,  j(e ") = e " a  T(e  C T  JL(e  C T  a  +  i(e ) +  a  and  ) = e ", i ( e CT  +  s.t.  j  O j  2: i  n  o' = ( 0 , 0, . . . , 0,  ) = e j , or  = e£,  l(e ") = e a  + C T  a  >  f o r a'  as above  e CT  from above, depending on cr.  The s t a t e m e n t s about from t h e d e f i n i t i o n s .  a t  ) = ^(e^)  e,  , a^, . . . , o^)  -T  k' j(e  k j n  T h e i r a c t i o n s on c e l l s a r e  CT  l(e ) = e i C T  and  n  j , j , 1, 1, T  The s t a t e m e n t s about J_  and  are easily v e r i f i e d  and _L a r e n o t so easy t o  v e r i f y — o n e way i s t o go t o graph c o o r d i n a t e s — b u t s i n c e t h e y a r e n o t used i n any i m p o r t a n t way, t h e y w i l l n o t be p r o v e d h e r e .  1.11 the  Remark:  By 1.10, t h e c e l l s t r u c t u r e s f o r Gj,,^ and  union l i m i t  yield  CW-complex s t r u c t u r e s f o r G^  CW-complex s t r u c t u r e s f o r  G  and  G.  and  G^  (The CW  n  yield i n  which i n t u r n p r o p e r t i e s a r e easy  t o check, e.g., t i l page 79.)  1.12 given.  Definition:  L e t CT = (cr^, . . . , cr^ - n ) ,  a Schubert s y m b o l , be  2(a)  i ) The u n o r i e n t e d Schubert v a r i e t y closure of  e  i n G^  CT  ^ (°,(cr)) = t h e c l o s u r e o f  e  _1  Remark:  i ) 2(a)  U e ~  + C T  in G  2(a)  i n G^  2(a)  i n G^  CT  is  n  .  n  k-planes  P fl j ( R  )  1  i s the set of oriented  n  i n G^  i s the set of  k  s a t i s f y i n g t h e c o n d i t i o n s dimension o f and  i s the  n  .  n  i i ) The o r i e n t e d Schubert v a r i e t y  1.13  in  -  P  in  R  k + n  i i = 1, . . . , k  k-planes i n R  s a t i s f y i n g t h e same c o n d i t i o n s . i i ) i f n' > n  t h e n B ( n , n , . . . , n) rv rv rv in ' = j(G] ).  *M  and  °.(n, n , . . . , n)  n  i i i ) i f k' 2: k  i n G^  k  i  j  is  n  l(G  k  n  )  and  then  8(0,  .  t = jCG^^)  < ) n  °.(0, 0  0  ,  n, n, k  G  n  . .  , 0,  n,  n,  ~ ~ K k,n)-  k  . .  n) i n times  . , n)  in  G^,  is  n  times  G  i v ) Suppose then and  j : G\,^  n  S2(CT) i n G^ i  Q(o')  n  ->• G ^ i  i n G^t  and  n' ^ n.  i n d u c e s a homeomorphism  n  n  i n d u c e s a homeomorphism where  Claim:  be g i v e n .  v  and pf:  2(a)  in  &(o~) i n G^  n  n  a^, . . . , o ^ ) .  1  T  - k  case.  L e t CT = ( c ^ , . . . , cr 5 )  Then  between  cr' = ( 0, 0, . . . , (3, o , k  Similarly i n the oriented 1.14  between  and  n  1 : G^ and  Gy-^i  k' > k  k  n  and CT' = ( c ^ ' , . . . , CT^' 5 n)  2(a) c S(CT')  in G  v  _ «=»CT-5CT.' V i  2(a) c 2(a')  in G  v  „ » CT-  5 CT-' V i .  U s i n g 1.13 ( i ) :  Suppose CT and  cr' a r e Schubert symbols as above and i - °i' V i . CT  P € 2(a) => dimension o f  CT.+i P fl j ( R ) > i  V i  Then  10 cr!+i P fl ] ( R ) > i  => d i m e n s i o n o f Thus  P € 2(a'),  =» P €  ffi(a )  f o r some  a  i.e., for a  i  5 a '  1  1  V i , P €2(a)  i  and a ' i^  .  Then  a r e Schubert symbols as i n t h e c l a i m , and CT^ > 2_^ <J  dimension o f  „ a,« +i -l P fl j ( R ) = i_ i  0  rt u  =>  dimension o f  P D j(R  0  U  P i 2(a'),  =» P ? 2 ( a ' ) ,  1  P € e^ c 2 ( a )  =»  Thus  ^ a.+i ^ a!+i j(R ) c J(R ).  since  o r 2(a) c 2(a<).  f  Suppose  V i,  1  n  ) 5 i  i.e., P € e  Q  Q  - 1  - 1,  since  j(R  ) c j(R  0  1 0  u  ) .  c 2(a)  o r 2 ( a ) <Z 2(a»).  The same arguments work f o r . t h e o r i e n t e d c a s e . 1.15  Remark:  We can c o n s i d e r a l l t h e Schubert v a r i e t i e s as f i n i t e  d i m e n s i o n a l subcomplexes  o f t h e u n i v e r s a l complexes  i n c l u s i o n maps  and  j ,j ,1  1  G  or  G,  since the  a r e homeomorphisms on any Schubert v a r i e t y .  The i n c l u s i o n s between t h e Schubert v a r i e t i e s can be shown by a diagram which i s c a l l e d t h e Hasse diagram.  The diagram i s v a l i d f o r b o t h o r i e n t e d  and u n o r i e n t e d Grassmannians. Diagram 1.16 shows a l l Schubert v a r i e t i e s l y i n g i n G^ ^ d i m e n s i o n 8.  up t o  Note t h e h o r i z o n t a l s y m m e t r y — i t r e f l e c t s t h e map J_  cellwise  There i s a l s o a v e r t i c a l s y m m e t r y — t h e o t h e r h a l f o f t h e diagram f o r Schubert v a r i e t i e s i n G^ ^ 8.  can be o b t a i n e d by r e f l e c t i n g a c r o s s d i m e n s i o n  T h i s comes from P o i n c a r e D u a l i t y .  Graph C o o r d i n a t e s and Chain Complexes f o r t h e Grassmannians 1.17  Definition:  Graph c o o r d i n a t e s f o r  F i x t h e s t a n d a r d b a s i s on  R ' . k  D e f i n e t h e graph c o o r d i n a t e s c e n t r e d a t let  G^  n  P  n  .  Let P  be a  as f o l l o w s :  P"*~ be t h e o r t h o g o n a l complement o f  P  in R k n +  k-plane i n  R ' . k  n  11  12 and h : P x P*- -»• R h ( v , w)  t h e isomorphism  k + n  = v + w  (vector addition)  Define cp  p  :R  fa Hom(P, P ") -* G  k X n  by  1  k > n  <P (f : P •+ P ) = h ( g r a p h ( f ) ) ,  a  1  p  k-plane i n  R  k + n  i ) T h i s g i v e s t h e graph c o o r d i n a t e s c e n t r e d a t P ii) If P (Pp : R  G  k x n  k j n  as above, g i v i n g  P & graph(f) ^ h(graph(f)). o  G  r  1.18  k j n  <p (f : P -> P ) 1  p  v i a t h e isomorphisms  T h i s g i v e s t h e graph c o o r d i n a t e s c e n t r e d a t P  .  Remark:  I n t h e graph c o o r d i n a t e s c e n t r e d a t P  c h o i c e o f isomorphism ordered b a s i s  Hom(P , P CT  )  CT  R  { a ^ + l ' ®cr +2» • • • » ^o" +k} e  2  f :P  CT  ->• P  a  n  d  k  §i  Then w i t h t h i s correspondence,  Give PQ-  v e  t  P^ t h e n  e  ordered  ( i n i n c r e a s i n g order a l s o ) .  k + n  correspond t o the matrix o f f  CT  , there i s a n a t u r a l  a  as f o l l o w s .  basis o f the remaining b a s i s vectors i n R Let  .  k n  i s an o r i e n t e d p l a n e , d e f i n e  the o r i e n t a t i o n induced by t h e o r i e n t a t i o n o f P  f  for G  i n t h e above b a s e s .  t h e map <P (= <Pp ) : R  G  CT  k n  (or G  k  n  )  i s g i v e n by A = ( a ^ j ) -> t h e k-plane w i t h r e p r e s e n t a t i o n ( s e e 1.2)  a  a  ll  12 ' ' ' l a  1  21 22 • • • 2 o  1  a  a  a  •  a  •  0  a  a  • • • ia  2  a  2o +l  • • • 2cr  2  2  a  0  a  a  a  2a +l  a  2  •  ka +l k  • • • ka a  0 2  lo +l  • • • lo" a  2  1  *  0 k  lo +l 1  •  l*kl k2 ' • ' ka a  1  a  0  • • • 2o a  a  ic +l  • • • l a  k  0  a  2o" +l k  k  •  n  • • • 2n a  •  • • •  ko +l • • • ko a  2  0 k  1 k  ^o^+l • • • kn a  13 and  (P (R  k><n  CT  )  i s t h e s e t o f p l a n e s h a v i n g such a r e p r e s e n t a t i o n . (The  above i s v a l i d f o r  G  Notation: Give A21,  Call  R  . . . ,A  k n  }  also.)  k n  cp (R  k x n  a  )  U  in G  0  the ordered b a s i s  k j n  {A^, A  and U  » • • • »  A  1 2  A^j = matrix with 1 i n i j  where  in G  + CT  t  n  l n  .  k ? n  ,  p o s i t i o n and z e r o s  everywhere e l s e .  1.19  Claim:  L e t o = ( o ^ , . . . , o" _ n) be a Schubert symbol. k  i) e <  A  11' A  in  R kxn  CT  c U  i n G^  ct  k  and cp (e ) Q  n  » A-lnr. > A 0  A  , An ,  Q  Then  i s the plane  . . . , A-  , A„. , . .  1 2 ' * " * ' "l "^' " 2 1 ' 2 2 ' ' * • » "202' " 3 1 ' * * * »  kl'  ? 1  00  • ' • ' ko > A  k  C a l l t h i s plane i i ) In G  k > n  , e  L^ . CT  c TJ  CT  and cp~ ( e  + a  ) i s t h e same p l a n e  L  Q  above. pf: e  CT  I t i s easy t o see t h a t a p l a n e  P €G  k  n  is i n  » i t has a r e p r e s e n t a t i o n o f t h e form  1 0  . 0  0 0  0 *  . *  10  0 0  0 *  0 0  . " 0 'CT 1'  1 0  0 0.,  0 *  * 1  1+  column  column  <=» P = <p (A) f o r some m a t r i x CT  a j_ j = 0  column A = ( a i j ) such t h a t  f o r i >CTJ_+ i + 1 .  T h i s proves p a r t ( i ) . P a r t ( i i ) i s proved t h e same way, n o t i n g t h a t lies i n e ^ .  P  a  14 1.20  Definition:  L e t CT = (cr^, . . . ,  e  D e f i n e the o r i e n t a t i o n o f i n d u c e d by  L  ,  CT  where  in  Q  L  CT  G  k  1.21 and  e ~ CT  via  e  be a Schubert in  a  G^  t o be t h a t  n  i s g i v e n t h e o r i e n t a t i o n determined by t h e C a r r y over t h e o r i e n t a t i o n o f  i ) We now have c e l l s t r u c t u r e s o f o r i e n t e d c e l l s f o r  j ,j ,1  and  1  (also  and  T)  orientations i n  Definition:  G^  , G^  , G  and  n  dimension  r.  i ) D e f i n e the graded group  Define  C(G ) k  and  C(G)  i i ) D e f i n e t h e graded group g e n e r a t e d by a l l the Schubert c e l l s Define  C(G ) k  and  C(G)  cell  G.  C(G  f r e e a b e l i a n group g e n e r a t e d by t h e Schubert c e l l s  Remark:  k  a l l preserve  the c e l l o r i e n t a t i o n s , so t h a t t h e s e c e l l o r i e n t a t i o n s induce  r.  G  .  k ? n  i i ) The maps  1.22  symbol.  T.  Remark: G  and  n  o r d e r e d b a s i s v e c t o r s which span i t . to  5 n)  e  )  as  C (Gj  CT  in  G^  r  n  <  n  )  = the  of  similarly. C(G  e  k > n  + C T  k ? n  and  )  as the f r e e a b e l i a n group e ~ CT  in  n  of  dimension  similarly.  These graded groups are t h e b a s i s o f c h a i n complexes f o r t h e  Grassmannians a r i s i n g from t h e o r i e n t e d c e l l  decompositions.  15  PART I I - ADDITIVE HOMOLOGY STRUCTURE  I n t h i s s e c t i o n , t h e a d d i t i v e s t r u c t u r e o f t h e i n t e g r a l homology of  G^  and  n  G^  w i l l be s t u d i e d by computing d i r e c t l y from c e l l  n  o r i e n t a t i o n s t h e boundary homomorphism . d a r i s i n g from t h e Schubert c e l l The f o r m u l a f o r  d  f o r t h e c h a i n complex  (G^ ; n  Z)  decomposition. (Theorem 2.9) i s t h e main r e s u l t aimed f o r ,  and t h e n some low d i m e n s i o n a l homology groups f o r Gj,.  n  and  G^.  are  n  derived.  G e n e r a l Theory f o r C e l l  Complexes  In g e n e r a l , g i v e n a for  each c e l l i n C  making  K,  CW-complex  t h e r e i s a homomorphism  f o r any group  A  triangulating  K  G.  d : C ( K ; Z) ^•C ~"''(K; Z) 1  Definition:  ( C , d) ® G  is  so t h a t t h e c l o s u r e o f each c e l l i s a f i n i t e subcomplex,  Let  K  be a  r e s u l t i n g boundary homomorphism. r - 1  1  ( T h i s can be done, f o r example, by  and u s i n g s i m p l i c i a l methods t o d e f i n e  and  t o g e t h e r w i t h an o r i e n t a t i o n  i n t o a c h a i n complex so t h a t t h e homology o f  H ( K , G)  2.1  K  d,  see [6].)  CW-complex w i t h o r i e n t e d c e l l s , and For  e  Q  and  e^  c e l l s of dimension  r e s p e c t i v e l y , d e f i n e t h e i n c i d e n c e number  ep-coefficient of  d  [e^, e ] a• R  a>  e  pl  =  0  o r  ±1>  dea •  which t h e f o l l o w i n g f a c t s w i l l t a k e c a r e o f :  r  t o be t h e  For t h e o r i e n t e d Grassmannians, t h e o n l y p o s s i b i l i t i e s w i l l be [e  the  16 2.2  Let  K  be a  o f dimension  r  CW-complex o f dimension and  i) If  e  r - 1  then  n  and  L  a linear  r  -  a  a  oriented  cells  Q  U  fl U, ep fl U)  +  e , ep  [ e , ep] = 0  i i ) I f t h e r e i s an open s e t <p : ( R , H , L) •* ( U , e  and  respectively.  D ep = 0  Q  n  in  k and a homeomorphism  where  l space bounding  H  H ,  i s a linear  +  r-half  space  then  +  r [e ,  1,  ep]  a  i f the o r i e n t a t i o n of  <  coincides by -1  i i i ) Let  e  e^  H  i n d u c e d by  +  e  a  with the o r i e n t a t i o n of  L  induced  f o l l o w e d by t h e normal o f  L  in  H  otherwise.  a l s o be an o r i e n t e d  c e l l o f dimension  r,  and  suppose t h e r e i s a homeomorphism <p : ( R , H , H", L) •+ (U, e n  where  L  space.  i s a linear Give  Then  +  H  +  [ e , ep] = Q  r  and  -  H-  l space and  Y  '  e  pl  y  n  U, e  H = H  +  fl  p  U)  U L U H~  induced by  e  i s a linear and  a  e  r  respectively,  i f t h e r e i s a change o f o r i e n t a t i o n a c r o s s L  e  U, e  fl  orientations  [e^, ep]  •t  a  in  H  otherwise  These f a c t s w i l l not be proved h e r e , but can be checked by g o i n g to a s i m p l i c i a l d e f i n i t i o n of  2.3  If k  and  f : k -+• k'  k'  are  #  i s a c e l l u l a r c o n t i n u o u s map t a k i n g  : C ( k ) •+ C ( k ' )  and i f f  ( s e e , f o r example,  i  i  6 ).  CW-complexes w i t h o r i e n t e d  p r e s e r v i n g o r i e n t a t i o n s , then f  d  f  r  c e l l s , and cells to  i n d u c e s a c h a i n map  Vi- (i.e.,  f  #  o d = d ° f )  i s s u r j e c t i v e o r i n j e c t i v e t h e n so i s  #  fg.  r  cells  T h i s i s a v e r y weak form o f t h e n a t u r a l i t y o f t h e c h a i n complex q(k)  D e t e r m i n i n g the I n c i d e n c e Number f o r t h e Boundary Map f o r  C-(G  )  V  The manner i n which t h e g e n e r a l t h e o r y i s a p p l i e d t o G, K  is ,n  b e s t e x p l a i n e d by an example. 2.4  Example:  The boundary map i n C ^ ( G  '•  2  R a t h e r t h a n use t h e c e l l o r i e n t a t i o n s g i v e n i n 1.20, c o n v e n i e n t t o d e f i n e t h e o r i e n t a t i o n s as we p r o c e e d . coordinates centred at ^ ( 0 0) l l 1 2 a  a  =  p  ( Qo ) '  h  w  e  r  it is  C o n s i d e r t h e graph  e  " ^ ( o r i e n t e d ) row space o f t  ie  *21 2 2 a  1 0 a  11  a  12  O l a  21  a  22  I t i s . e a s y t o see t h a t t h e c e l l (0,  1)  +  corresponds t o t h e  l i n e a r h a l f space (recall  <  A2i | >  a  A ^ j from 1.18), and  ( o ,  1)~  c o r r e s p o n d s t o <A >  ( o ,  2)  +  corresponds to  (1,  D  +  (1,  D ~ corresponds to  <A  ( o ,  2)" corresponds to  <A , A  21  a2i<0  <A , A >| 21  Fig. 1  2 2  corresponds to < A , A i > | n  11» 21  2  A >| 2 1  2 2 > |  a^-]>0 a <0 22  G i v i n g t h e s e l i n e a r h a l f spaces o r i e n t a t i o n s , i t i s easy t o determine from them t h e i n c i d e n c e numbers [ ( 1 , 1 ) , ( 0 , 1 ) ] , +  [(1,  1 ) , (0, I ) ] , +  -  e t c . and t h u s o b t a i n  +  d ( l , 1 ) , d ( l , 1 ) " , d(0, 2 ) +  +  18 and  d(0, 2)~.  I n o r d e r t o determine  c o m p l i c a t e d procedure  d(l,2)  i s needed s i n c e  (1, 2 )  i n these coordinates ( i t i s t h e s e t  l l  a  |a  a  go t o new graph c o o r d i n a t e s where t h e c e l l (1,  1)  and  +  (0, 2 )  (P^  +  ^  Q  d ( l , 2)~,  A :  2 2  a more  i s n o t a l i n e a r h a l f space  +  12 a  2 1  and  +  I At  We must  = 0  j  (1, 2 )  +  i s l i n e a r as w e l l as  w i l l w o r k ) , and keep t r a c k o f t h e  o r i e n t a t i o n s induced by t h e c e l l s  (1, 1 ) , (1, 1)~, (0, 2 ) +  +  and  on t h e i r c o r r e s p o n d i n g l i n e a r subspaces i n t h e new c o o r d i n a t e s . the main t e c h n i c a l i t y i n t h e p r o o f o f P r o p o s i t i o n 2.8. i n a s i m i l a r manner o b t a i n  (0, 2 ) ~  This i s  Here, we can  d ( l , 2 ) , d ( l , 2)~, d(2, 2 ) +  +  and d ( 2 , 2 ) ~  which t o g e t h e r w i t h t h e above w i l l y i e l d t h e homology o f G  2  stated i n  2  Table I I . Note:  The boundary map c a l c u l a t e d above w i l l not n e c e s s a r i l y be t h e  same as i n 2.9 as t h e c h o i c e o f c e l l o r i e n t a t i o n s might be d i f f e r e n t . 2.5  Notation:  o - 6  S  F o r o~ = (CT^, . . . ., 0" )  t h e symbol  Kronecker  (CT - 6 1  Lemma:  , . . . ,  i s a Schubert  S  L e t cr = (CT^, . . . , |CT| = |CT'| + 1.  such t h a t  - 6  )  k g  where  6.  g  i  s  by  the  symbol  S n)  Then i n G  k  n  unless  » CT S CT - 1. S  and  S  cr' be Schubert  symbols  ,  [e*,  ej,] = [e*, e i] = 0  pf:  T h i s f o l l o w s from ( t h e p r o o f o f ) 1.14 and 2.2 ( i ) and t h e f a c t  that i f 2.7  l g  symbol, denote  5. Note: CT - 6  2.6  a Schubert  k  C T  cr' / cr - 6^, f o r any s  Lemma:  s  cr' = cr - 5  then  a. i  L e t o = (cr^, . . . , o^ 5 n)  such t h a t CT' = CT - 5  s  f o r some  s.  = cr! i  Then i n G  o r i e n t a t i o n s as d e f i n e d i n 1.20) we have  +1  0  and  f o r some  g  s.  f o r some  0  cr' be Schubert k  n  ,  (with c e l l  i„ . 0 symbols  19 i)  [e+, e j , ] = (-l)° - [e-, s+k  [e+, e+,] = (-1)  ii) pf:  e+,]  s  3  +  1  3  +  ^  2  T h i s i s proved by g o i n g t o graph c o o r d i n a t e s where t h e lemma  t a k e s t h e f o l l o w i n g form. P r o p o s i t i o n : Suppose CT and cr' a r e Schubert  2.8  symbols as above and  C | T'| = r . Then i n t h e graph c o o r d i n a t e s c e n t r e d a t P ' we have: t  i) e,  c  C T  c a l l t h i s plane ii) e ( c a l l them  H  and i s a c o o r d i n a t e  U i  a  As i n 1.19  Xn  L, . Q  + C T  fl U  C T  i  and e ~ (1 U » a r e c o o r d i n a t e  and H~)  +  r-plane i n R^ .  such t h a t  H  +  U L , U H~  r + 1  h a l f planes  i s a coordinate  r + 1  plane. i i i ) The o r i e n t a t i o n s o f H  +  and H  induced by  e  + C T  and  CT  e  a r e t h e same » o" + k - s s  i s odd.  i v ) The o r i e n t a t i o n o f H o r i e n t a t i o n o f L_  +  induced by  e  + C T  coincides with the  ( a s d e f i n e d i n 1.20) f o l l o w e d by t h e normal i n t o  H  +  <=» k - s + CTG+1 + CTS+2 + . . . + CT^ i s odd. Note:  The above, t o g e t h e r w i t h 2.2 ( i i ) and ( i i i )  proves 2.7 pf:  U s i n g t h e above n o t a t i o n : i ) T h i s i s 1.19. ii) P € e  + C T  fl U , CT  » i t has r e p r e s e n t a t i o n s (see 1.2 and 1.18) o f t h e forms  immediately  20  X  l  =  * 1  0  0  0  0  *  0  0 . .  row s  ft o  1  0  0  *  *  * 1  0*^+1 column  o  V s column column s-1  CT  +s  k +  0 k  column  and  1 0  row s  ft  o ft  crj+1  CT'+S  =CT +1  -O+s-1  s  1  o'+s+l s  ft 1 ft CT'+k k  =cr„+s  =a k  S  column  k +  column  column  column  » t h e r e i s an o r i e n t a t i o n p r e s e r v i n g l i n e a r t r a n s f o r m a t i o n representation of > 0  i n the  the columns  k x n  X  representation  2  + i  V i , and  X  2  cr  ( T h i s can be seen by l o o k i n g o n l y a t + s -  1.)  r e p r e s e n t a t i o n w i l l have  O's  t o the r i g h t of the  e x c e p t f o r row s w h i c h w i l l have a p o s i t i v e number  CT + s, s  t o an  representation.  In such a c a s e , t h e l's  P  and matrix  O's with  t o the r i g h t .  t a k i n g the  Thus  P = <P^ (A) where t  (1/©)  i n column  A = (a^j) i s a  21  f o r j > a-  a,--: = 0 -L J  Conversely, forms  and  a,, ^  X^  and  X^  t> ,  X  and  has r e p r e s e n t a t i o n s o f t h e  .  2  P  > 0.  U o  A , P = cp^,(A)  f o r any such m a t r i x  Similarly, form  + i + 1  _L  P € e ~ fl U_, » T ( P ) a  has a r e p r e s e n t a t i o n o f t h e  has a r e p r e s e n t a t i o n o f t h e form  X  2  «• t h e r e i s an o r i e n t a t i o n r e v e r s i n g l i n e a r t r a n s f o r m a t i o n t a k i n g t h e r e p r e s e n t a t i o n t o an ~ © < 0 » P = for  <P  C T  X  representation  2  where  ,(A)  i _ CT l• + i + 1  and  J  X^  A  = (a^j)  isa  k x n  matrix with  a^j = 0  a„ _ > 0. S,CT S  This proves ( i i ) . From ( i i ) we have t h e diagram shown.  F o r ( i i i ) and ( i v ) we must f i n d  the o r i e n t a t i o n s o f induced  by  e  + C T  ,  (H  and  +  H  H  of  o f t h e maps  -1 cp o T o cp^, .  and  CT  and  which can be done by  f i n d i n g the Jacobian -1 <P- ° <P t  H  CT  have n a t u r a l  o r i e n t a t i o n s g i v e n by t h e i r o r d e r e d  <A1 1 '  A  1  ' ka >' A  k  induced _i  by  o cp ,  CT CT  e  + C T  The o r i e n t a t i o n  w i l l be t h e same i f Fig.  O  ° T ° <p t C  2  has J a c o b i a n w i t h p o s i t i v e  d e t e r m i n a n t , and t h e o r i e n t a t i o n induced cp  coordiyvxV«  ' la ' 2l' A  'kl'  cp  basis  has J a c o b i a n  by  e^  -  w i l l be t h e same i f  with p o s i t i v e d e t e r m i n a n t . ) To w r i t e t h e s e maps  T  c o o r d i n a t e w i s e , we must see how t o go from one r e p r e s e n t a t i o n t o a n o t h e r . Given  P € U  CT  fl U _ i ,  t y p e r e p r e s e n t a t i o n f o r P.  l e t v^, . . . , v  k  be rows o f t h e  To o b t a i n a r e p r e s e n t a t i o n o f t h e form  X  2  ,  22 use rows way:  w, 1  w r i t e t h e **s  (A = ( a - ) v  s  / a  s,o  a  n  where t h e w-'s 1  -1 <P ,(P) a  w  d  s  k  in X  w i l l be  i:  w„  . . . ,w  £  as  a^.  and  a  i n t h e a p p r o p r i a t e manner w i l l be © — s e e  s > Q  = vj_ - ( a i C T/'a" s o -) ( "sv ) i C T  /v  s  s  Note:  t h e determinant  1.18).  Then  i * s  s  s  of t h i s transformation i s l / a s o  thus i s o r i e n t a t i o n p r e s e r v i n g f o r a scr  are obtained i n the f o l l o w i n g  > 0  S C J  , and  and o r i e n t a t i o n r e v e r s i n g i f  s  Working out t h i s l i n e a r t r a n s f o r m a t i o n i n c o e f f i c i e n t s , we g e t cp;*(P) = ( a  l m  ) - ( b . J = cfTV)  ~ io-  W  for  i = s,  for  i  for  i = s,  a  here  j  i s, j  = CT  so j f cr  ser a . • a. s] ICT  for  scr„  2.8a. and  F o r convenience c a l l t h i s map f a so  s  determinant  < 0 is Tocp.ocp cr cr  i / s.  f.  Then  fij(a  l m  ) = bij ,  i cr  f l a scr  > 0  We must now  o f t h e J a c o b i a n when we r e s t r i c t  For  n  f  is  •1  ip" o m  f i n d the to  L  .  the p a r t i a l d e r i v a t i v e s are  3f. 3a7~  =  6  i j °jm  for  1 1 s,  m i o  s  , i i- s  and  j f  CT  Thus r e s t r i c t i n g t o these c o e f f i c i e n t s g i v e s us t h e i d e n t i t y  23 m a t r i x so f o r t h e d e t e r m i n a n t we need o n l y worry about t h e k - s  - s + cr s  + c r x k s  matrix  3f where  3a lm  and  i = s m 5  or  j = a, ,  1 = s  or  m = a  ,  s  and  j < a.  CT-^  This matrix i s S r  3f  si  a  sl  3 a  3a  s2  -1 ( aSO_~_ ) (a  ...  3a, „ ko s  0  -1  ->  af s2  3a„,., _ S+1CT s  so\ s  s  af so  ( a „S0\, „ )  af S+lCT  -(a_  -2 ) •(a_ ) S0  0  af ko the  2.8b.  The d e t e r m i n a n t o f  J f| |  2.8(iii). on  H  and  H  o~ + k - s  '  *  *  J  i s thus  T  column  (-l) ~ k  s + 1  )/(a  C T  o\,+k-s+l )  o r r  E  S  From 2.8b and 2.8a above we have t h a t t h e o r i e n t a t i o n s  agree  2.8(iv). »  J  3a„^ scr„  -1  i s odd.  s  <A 11  0  '  A  Comparing f i r s t t h e o r i e n t a t i o n  lCT '' "? 21 1>'  1CT.  A  1  -  •  '* '' '' " 92r cr _ '> •• •• *• '> " k lH >> * * ' ' ko\ ^ k A  A  A  >  o  f  H  with the orientation <A  A  11> k l  , A  • • • > l o » 21> A  A  1  , . . . , A  k a  >  of  L^,  s l  ,  . . . , A  f o l l o w e d by  A go  s C T  ^_  l S  A  g + 1  ,  .  ( t h e normal i n t o  H ) +  we  have agreement CT  s+l  +  CT  s+2  +  + Oy. i s even.  Comparing t h e above o r i e n t a t i o n f o r  24 H ,  w i t h t h a t i n d u c e d by  +  » (-l)  i s odd.  k - s  —• k - s + o " 2.9  +  s+1  Theorem:  CT  e*,  by 2.8b we have agreement  Combining t h e two, we have agreement 2  • • •  +  s+  +  k  CT  The boundary map  ^  Q-E.D.  s  d  i n t h e c h a i n complex f o r  G  k  with  n  c e l l o r i e n t a t i o n s as i n 1.20 i s ,, +  d(e_)  V  =  x  / i xl+k-s+o <i + . • .+Ov/  I (-1) s s.t.  s+1  c:+  (e  k  + C T  _  / „ k-s+o  + (-1) x  5  _  e _  s  a  s  )  5 s  and  d(e~) = Td(eJ) pf:  T h i s f o l l o w s d i r e c t l y from 2.7 and 2.8 and t h e f a c t t h a t  T  preserves c e l l o r i e n t a t i o n s .  2.10  Corollary:  The boundary map  d  i n t h e c h a i n complex f o r  G  k  n  with  c e l l o r i e n t a t i o n s as i n 1.20 i s .  v  d(e_)  pf: e*  2.11  and  =  1+k-s+CT + . . .+o+ 1  I (-1) s s.t. CT ,5CT -1 s-1 s  3 + 1  k-s+cr„  k  K  ( l + (-1)  CT  to  Remark:  e  CT  F o r j , 1, j  k  C T  .  5  >|r : G  k  n  G  k  n  maps  preserving orientation.  c h a i n maps commute w i t h are v a l i d i n G  )e _  s  T h i s f o l l o w s from 2.9 and t h e f a c t t h a t e  3  and  d §  k  and  1  t h e embeddings i n 1.4, t h e i n d u c e d  (from 1.21 and 2.3) so t h a t t h e above f o r m u l a s ,  and a l s o i n G  and  G  ( i f we t h i n k o f each  Schubert symbol CT as s t a r t i n g w i t h a nonzero i n t e g e r problem o f h a v i n g an i n f i n i t e number o f CT^).  CT^  t o avoid the  25  Some Low D i m e n s i o n a l Examples  F i n d i n g t h e homologies o f t h e u n o r i e n t e d and o r i e n t e d  Grassmannians  and Schubert v a r i e t i e s reduces v i a 2.9 and 2.10 t o a l g e b r a i c c o m p u t a t i o n which w i l l be c a r r i e d out over  Z  i n some examples below.  In general,  homology o v e r o t h e r groups can t h e n be determined u s i n g t h e U n i v e r s a l C o e f f i c i e n t s theorem, b u t i n t h e f o l l o w i n g case i t i s e a s i e r t o compute t h e homology  2.12 for  d i r e c t l y from t h e c h a i n  complex:  H (G; Z ) = C (G; Z )  for a l l  Theorem: G^.  and  pf:  r  2  G^^  for a l l  r  2  k  and  r , and t h e same i s t r u e  n.  From 2.10, t h e boundary map  d  i s 0 mod 2  i n a l l dimensions.  The method used i n t h e examples i s t o f i n d i n d i m e n s i o n o f f r e e g e n e r a t o r s f o r t h e group o f c y c l e s (denoted boundaries homology  d(C ^) p  = Z /B r  a set  Zp) and w r i t e out t h e  (denoted B ) i n terms o f t h e s e g e n e r a t o r s . The  r +  H  r  r  r  i s then the s e t of generators of  Z  r e l a t i o n s g i v e n by s e t t i n g t h e boundary elements t o z e r o .  together with  r  The main  d i f f i c u l t y i s i n looking f o r a s e t of free generators f o r Z  ,  r  as i t i s  not always c l e a r whether o r n o t a s e t o f c y c l e s spans t h e whole o f  Z  p  ( a l t h o u g h t o s i m p l i f y t h i n g s , l i n e a r independence i n t h e examples g i v e n i s o b v i o u s , and i t i s easy t o determine what t h e rank o f  Z  r  should be).  In  a l l t h e cases worked o u t , t h e above p o i n t has been s e t t l e d by i n s p e c t i o n , which i n h i g h e r dimensions i s n o t p o s s i b l e .  2.12a (where  Note: \a\  In  G,  = r + 1)  , when w r i t i n g boundary elements i n terms o f g e n e r a t o r s f o r Z  then we need o n l y worry about  da . +  r  ,  do  and  i f da  +  da  = ±da~  Thus i n w r i t i n g t h e b o u n d a r i e s i n terms  26 of generators  of  Z  ,  r  some Schubert symbols y i e l d two e x p r e s s i o n s and  some o n l y one.  In Table  I  and i n t h e examples 2.15, a s h o r t e n e d n o t a t i o n w i l l  be used. 2.13  Notation: i ) Any Schubert symbol CT = (CT^, . . . , CT^) w i l l be w r i t t e n  a^ov,  . . . ay.  ( a s ay 5 9  i n a l l cases, t h i s w i l l not give r i s e t o  c o n f u s i o n ) and l e a d i n g z e r o s w i l l be o m i t t e d .  The z e r o symbol w i l l be  denoted i i ) In  G,  t h e symbols  +-o~  w i l l r e f e r t o a l i n e a r combination  +-23  CT  23 + 14 (2,  and  —CT)  and t h e second s i g n t o t h a t o f  +  ;  cr ; +  r e f e r s t o t h e c h a i n element (2,  ++a  cr w i t h one ( o r no) s i g n a t t a c h e d t o i t w i l l r e f e r  CT~. A Schubert symbol  e.g.,  -+a,  o f a n t i p o d a l Schubert c e l l s where t h e  f i r s t sign r e f e r s t o the c o e f f i c i e n t of  to the p o s i t i v e c e l l  (similarly  3 ) - (2, 3)' +  r e f e r s t o t h e c h a i n element 3 ) + (1, 4 ) +  +  TABLE I : THE BOUNDARY MAP I N 1 G  2,2  G  2,3  G  2,»  G  2,5  G  3,3  G  3,5  1-*-+*  2 2-+--1 11-*—1  3 12-*—11 ++ 2 3-*-+ 2  4  13-*-+ 12 — 3 4-*— 3  112-)— 111 +- 12  C ( G ) : (NOTATION FROM 2.13, "-*» REPRESENTS d ) .  5  23-*-+ 22 -+ 13 14-*— 13 ++ 4 5-*-+ 4 113-*-+ 112 -+ 13 122-*+- 112 -+ 22  6  7  33-*—  23  24-*— +15-*-+  23 14 14 5 122 113 23 122 113 14  123-*-+ -+ +222*— 114-*— +-  34-*— 33 ++ 24 25-*-+ 24 -+ 15 223-*-+ 222 + +123 133-»— 123 -+ 33 124-*— 123 +- 114 -+ 24 115-*-+ 114 -+ 15  8  44-*+-  9  10  34  35-*-+ 34 — 25 233-*— 223 — 133  45-*-+ 44 -+ 35 333-*-+ 233  134-*— ++ +224-*— — 125-*-+ -+ +-  144-*+-+ 234-*— ++ ++ 135-*-+ — -+ 225-*-+ ++  133 124 34 223 124 124 115 25  134 44 233 224 134 134 125 35 224 125  1111-*--111 1112-*—1111 1113-*-+1112 1114-*—1113 1124-*—1123 1134-*—1133 ++ 112 — 113 ++ 114 +-1114 ++1124 1122-*+-1112 1123-*-+1122 — 124 ++ 134 — 122 -+1113 1133-*—1123 1233-*—1223 ++ 123 —1133 . — 133 ++ 233 1222-*—1122 1223-*-+1222 ++ 222 ++1123 1224-*—1223 —1124 — 223 ++ 224 2222-*+-1222 2223-*-+2222 -+1223  55-*—  45  244-*+— 334-*— +-  234 144 333 234  145-*-+ 144 -+ 135 +- 45 235-*-+ 234 — 225 — 135 1144-*+-1134 — 144 1234-*—1233 ++1224 ++1134 — 234 1333-*-+1233 — 333 2233-*—2223 +-1233 2224-*—2223 +-1223  28 2.14  Theorem: i)  ;  H (G r  ii) H (G r  l 9 n  =4  Z)  Z  for r = 0  0  otherwise  =LZ  , Z)  for r = 0 if  Z  and  and r = n  i s odd  for r < n  2  ^0  n  n  and odd  otherwise.  pf: i) (r)  z r  + (r)  even.  (G  , Z)  l j n  for r  For r = 0  a r e g e n e r a t e d by t h e c h a i n elements  odd, r - n, Z  i s g e n e r a t e d by  p  e a s i l y seen from t h e T a b l e I.) (Image o f  d : C  r + 1  -> C )  + (r)  for r  (r)  +  - (r)  f o r 0 5 r < n.  r  = Z /B r  r  (0)  +  and  f o r n > r > 0 and ( 0 ) ~ . (This i s B  r  i s g e n e r a t e d by  p  +  H  (r) - (r)  The boundary group  (r)  Thus  and  odd  r < n  i s z e r o except f o r r = 0  and  n  where i t i s Z.  i i ) From 2.10, t h e f o l l o w i n g can be v e r i f i e d Z  r  g e n e r a t e d by  (r)  i  0 (0)  B  Thus  g e n e r a t e d by  Z /B r  r  is  2(r)  1° Z  r  odd  r > 0  even  (the 0-cell) r < n r  for r = 0  odd  even o r  r = n  for r < n  and odd  <Z  for r = 0  and  0  otherwise.  v.  2  r = n  i f n  i s odd  29  2.15  Examples:  I n t h e s e examples, T a b l e  f i n d generators f o r  Z  r  .  "a").  i s used by i n s p e c t i o n t o  I n l a b e l i n g t h e c y c l e s , no d i s t i n c t i o n i s made  between c y c l e s o f d i f f e r e n t dimensions labeled  I  ( e . g . , both  ++1  and +-2  are  As i t w i l l always be c l e a r what d i m e n s i o n i s b e i n g t a l k e d  a b o u t , t h i s s h o u l d n o t cause any c o n f u s i o n .  N o t a t i o n i s as d e s c r i b e d i n  2.13.  5  i)  0  Dimension: Generators for  Z  p  :  2 > 3  1  2  ...+ a=++l  a="  : b=*~  3  r  : a+b  a=+-33  a=++22  b=+-ll  b=+-12  b=+-13 +-22  +  Z. / B r  r  d(12)=  d(13)~=b-a  2c-a-b  d(22)=b  =  Iz r = 0, 2, 4, 0  dimension 2, generates  a=++23  a=++3  v.  c  6  d(2)=-a d(3)=-a d ( 1 3 ) = - b - a d(23)=-b d(33)=-a  I t i s easy t o see t h a t  (In  5  a=+-2  c=2-ll B  4  Z r  /B  Z /B r  r  otherwise  has t h e r e l a t i o n s  and has o r d e r  r  a = 0  2c  Thus  0.)  In t h e n e x t examples o n l y t h e f i r s t homology groups a r e d e t e r m i n e d , as t h e r e s t can then be found u s i n g U n i v e r s a l C o e f f i c i e n t s and Poincare d u a l i t y since  G  v  _  i s oriented f o r a l l  k, n.  i i }  5  3,3  Dimension ( r ) : Z  r  0 a=*  :  1 +  2  3  a=++l a=+-2  b=*~  a=++3 ,  b=+-ll b=+-12  4 a=++22 b=+-13  c=2-ll c=++lll  +-22 c=+-112 -+22  B  r  a+b  :  a  a  b±a  2c-a-b b  b 113: b+c  111: b 112:. b±c 122: c See note 2.8a.  r  V r B  Z  0  for  r  0, 4  for  r  2  for  r  1, 3  iii)  G  3,4  Dimension ( r ) :  3  Z : r  B  r  a ,b,c  :  4  5  6  a,b,c  a=++23  a=+-33  d=22+13-4  b=++113  b=+-222  e=+-4  c=++122  c=++123  d=122+113 -23  d=+-114-+24  e=+-14  e=++24—33  a,  34: e  4: a ,  b ,c ,b+c  b±a,b  14: 2d-b+e-a 24: a±e  124: c±d  114 b±e  133: c±a  222:±c  223: c±b  b±c  123: 2d-c -bta The u n e x p l a i n e d elements i n Z  and  r  B  a r e from Examples  p  ( i i ) and ( i  above, u s i n g t h e same l a b e l s . r  H  r  = Z /B r  r  =  <  0  r = 3  Z © Z  r = 4 r = 5  The homology groups  H  Q  , H  and  1  6.  and  H  2  a r e t h e same as i n G,  32 iv)  SKI, 4): 1  Dimension ( r ) : r  C  1  :  2 ll ;2  +  +  3  4  12 ;3  +  +  and  5  13 ;4  +  +  14  +  +  antipodals  a=++l a=+-2 b=+-ll  a=++3  b'=++13—4 e=+-14  b=+-12  e=+-4  c=2-ll  B  r  :  H : r  v)  11: a 12: 2c-a-b 13: b±a 14: b' 2: a  3: a  4: a  0  Z  0  2  3  Z  Z  2 ( 1 , 2, 3 ) :  Dimension ( r ) : C  r  :  1 1  +  ll ;2 +  +  4  12 ;3 ;lll +  +  +  5  13 ;22 ;112 +  +  +  6  23 ;122 ;113 123 +  +  +  +  and a n t i p o d a l s Z : r  a  a;b;c  a;b;  a=++22  a=++23  c=++lll  b=+-13+-22  b=++113  c=+-112-+22  c=++122  c =++123  d=122+113- 23 B  r  :  a  a;b;2c-a-b b±a;b;b±c  b;b+c;c  2d-c-b+a  H  r  :  0  z  z  Z©Z©Z  2  0  Z  33 vi)  G^ i| : a, b ,  Cycles:  Dimension 6  c, d, e  as i n Example ( i i i ) and i n a d d i t i o n ,  f = 1122 + 1113 - 114 - 222  +24-33  g = ++1113—114 and  h = ++1122—222.  Boundaries:  e , c ± d , c ± a , c ± b  1114:  g,  1123:  1222:  as b e f o r e , and i n a d d i t i o n ,  h  2 f - h - g - e ± c .  Thus i n homology we have e = g = h = 0, Thus  f  a = b = c = d, generates  H  2f=c  and  g  T a b l e s o f Homology Groups o f t h e  and  4 f = 0,  2c = 0. 2 f £ 0,  so  H  g  = Z^ .  Grassmannians  Table I I I t a b u l a t e s the above r e s u l t s t o g e t h e r w i t h a few more t h a t have been worked out i n the above manner. Grassmannians, III  .  such r e s u l t s a r e v a l i d f o r  G^  By g o i n g t o l a r g e enough and  G  as shown i n T a b l e  Cohomology can be found u s i n g U n i v e r s a l C o e f f i c i e n t s , and t h e r e s u l t s  can be compared w i t h t h o s e o b t a i n e d u s i n g c h a r a c t e r i s t i c c l a s s e s ( s e e [ l ] pages 179 and 182).  The c o p i e s o f  Z  a r e g e n e r a t e d by P o n t r j a g i n c l a s s e s  and t h e i r p r o d u c t s .  Another method would be t o use t h e c o c h a i n complex  d i r e c t l y , where the i n c i d e n c e numbers d e f i n i n g would be  [ e , ep] = [ep, e ] a  a  6  ( t h e coboundary  from t h e boundary map.  map)  Going through t h e  same procedure as i n Example 2.15, e x p l i c i t g e n e r a t o r s i n terms o f Schubert c e l l d u a l s c o u l d be d e t e r m i n e d .  I n t h i s way f o r i n s t a n c e i t c o u l d be found  which Schubert v a r i e t i e s r e p r e s e n t t h e P o n t r j a g i n c l a s s e s .  Table I I .  Homology groups f o r  5  2,2  o  H  H  l  H  2  H  3  <4 5  H  6  H  7  2,3  H  10  H  l l  g  2,B  S,3  3,<4  G  13  H  1H  H  15  H  16  n  are small  S,5  Z  Z  Z  Z  z  0  0  0  0  0  0  0  0  z  z  Z  Z  0  0  0  0  0  z  z  z @ z  Z  z  z ® z  0  G  0  z  Z  2  z  Z  z  Z  Z  2  0  0  0  Z  z  0  0  z  2  2  Z  Z  2  Z  2  0  0  0  z  z « z © z  2  . 2  0  \  0  z  0  ZfflZ  z  Z®ZfflZ@Z  2  Z  Z  2  Z  Z  2  z  4 2  0  0  Z  0  z  0  z  Z  z  12 H  and  z  8 9  2,4  k  Z  H H  G  where  Z  z e e  H  H  G  G,  2  z @ z ® z  0  Z  0  0  z  0  2  z  r  2.17  Assertion:  H (G r  2  n  ) = <Z  for Z © Z 0  The method used f o r  Corollary:  r  for  G  for  r  even-and  r = n  r 4 n,  r 5 2n  even  otherwise.  G, <2 2> • • • > 2 5  H (G2) = Z  r  even,  c a n  0  ^  e  for  generalized r  odd.  easily.  35  Table I I I . (Note: c e l l s of H (G r  r +  H r  Low d i m e n s i o n a l homology groups f o r G  (G ) =  H  k  G  r  H  5  l  3  %  (-2]<- i)  r + 2  H  l  2  3  \  H  5  H  0  0  0  z  0  z  0  z  0  0  z  0  Z  2  0  z  Z  z  0  Z  2  0  z ® z  z  0  z ® z  2  0  z ® z  2  2  5  6  z  0  Z  2  z  0  Z  2  z  0  z  2  •  0  k  covers a l l  2  0  2 z  2  z  ® z  2  2  Z ®Z  •  2  •  z ® z  -* G  6  0  Z  + 1  f o r t h e same r e a s o n . )  0  0  k  :  Also,  0  z  G  H  G  z  5  ?  or less.  r  S  G  r + 1  ) = . . . = H (G)  H  o  s i n c e t h e embedding  5r+  o f dimension  k  i) = H (G  G  r  K  2  2  z © z  2  © z  2  z  2  © z  2  © z  2  z  2  © z  2  © z  2  In t h e u n o r i e n t e d Grassmannians and Schubert v a r i e t i e s , t h e computations a r e much s i m p l e r as t h e r e a r e o n l y h a l f as many c e l l s t o worry about and t h e boundary map i s much s i m p l e r .  36  T a b l e IV. Homology f o r t h e u n o r i e n t e d  Grassmannians  G, K  k  and  f o r small jii  n:  G  2,2 Z  l  H  H  Z  2  3  H  \ H  5  H  6  H  2,3 Z Z  2  z2  z2  0  Z  z  z  Z  2  8  H  9  H  10  H  l l  G  2,4  G  2  0  2,5  Z Z  Z  2  z2 Z  z2  2  Z  2  Z  2  2  Z  2  Z  z®z2  y  H  H  2  G  z©z2 z2©z  '0 Z  2  G  3,3 Z Z  2  Z  2  z2©z  Z  Z  2  2  Z  2  Z  2  Z2*522 z„©z„  2 2 Z©Z„ 2  0 z  Z  2^2  0  0 Z  2 0  1 2  Note t h a t t h e r e i s P o i n c a r e D u a l i t y i n G This r e f l e c t s the f a c t that  2  z©z2©z2 Z  2  , 2  z©z2  2  Z  3,4  Z  z  Z  Z  G  , G ., and 2,2 ' 2,4 0  0  G j ^ i s o r i e n t a b l e whenever  0  k + n  G °3 3 ' s  i s even  (see [ 2 ] ) . Remark:  The homology groups f o r G  k > n  have been determined i n [ 7 ] , b u t as  t h i s a r t i c l e was n o t a v a i l a b l e i n R u s s i a n o r E n g l i s h i t was n o t p o s s i b l e t o compare r e s u l t s .  37  PART I I I - HOMOLOGY AND COHOMOLOGY PRODUCTS  We now t u r n t o t h e m u l t i p l i c a t i v e s t r u c t u r e s ,  Only t h e  Z  2  homology and cohomology p r o d u c t s i n t h e u n o r i e n t e d Grassmannians a r e s t u d i e d , but t h e cohomology r i n g s t r u c t u r e i s determined e n t i r e l y (3.16 and 3.17). The f o r m u l a s d e s c r i b i n g t h e cup p r o d u c t a r e e q u i v a l e n t v i a P o i n c a r e d u a l i t y ( d e s c r i b e d i n terms o f Schubert symbols in  Z  cohomology o f G^ ( C )  i n 3.7) t o t h o s e d e s c r i b i n g p r o d u c t s  ( s e e [ 3 ] ) . I n t h e form g i v e n , they can a l s o  n  be used t o determine cup p r o d u c t s i n t h e u n o r i e n t e d Schubert v a r i e t i e s , and some examples a r e g i v e n .  The' G e n e r a l I n t e r s e c t i o n Theory To Be Used  For  M  a m a n i f o l d , there i s a product theory f o r i n t e r s e c t i o n s  o f c y c l e s i n H ( M ; Z^) t  n  L  :  H  n-a  ( M  c a l l e d the Lefschetz i n t e r s e c t i o n  ' 2> * n - b Z  H  ( M  ' 2> - n - a - b ' 2> Z  H  ( M  Z  which i s r e l a t e d t o t h e cup p r o d u c t i n cohomology i n t h e f o l l o w i n g way. 3.1  Assertion:  For b = n - a  D : H _ (M; Z ) * H (M; Z ) n  a  2  a  above, t h e p r o d u c t fi^ i n d u c e s a map  Z  2  H (M; Z ) Q  2  which can be c o n s i d e r e d as a  map D : H _ ( M ; Z ) -* Hom(H (M; Z ) n  a  by  2  a  a -y  t h e map  2  Z ) w H (M; Z ) 3  2  £  f ( B ) = a fl  B€ L  Z  I  B*  afl B=l L  If  a n  L  p = Y  i n H (M; Z ) then A  D(a) u D(B) = D ( Y ) i n H*(M; Z ) . 2  T h i s i s due t o t h e L e f s c h e t z i n t e r s e c t i o n p r o d u c t b e i n g P o i n c a r e d u a l t o t h e cup p r o d u c t i n t h e sense t h a t t h e f o l l o w i n g diagram commutes:  38  (with (M)  x  coefficients)  2  a+b. . (M)  H  H"(  [Ml  n  Z  D  n  [M]  [M]  y  -*H \ (M) n-a-b  H _ (M) x H _ (M) n  where  <-> [M]  HQ(M; 3.2  a  n  b  i s t h e cap p r o d u c t w i t h t h e fundamental c l a s s o f  (lying i n  Z ) ) i . e . t h e P o i n c a r e d u a l map. 2  Assertion:' Let M  s t r u c t u r e , and and  M  B.  e  a  , e^  be an  n  d i m e n s i o n a l m a n i f o l d w i t h a c e l l complex  c e l l s i n t h e complex r e p r e s e n t i n g  Suppose t h e r e i s c o n t i n u o u s map  h : M -> M  Z  2  cycles  a  homotopic t o t h e  i d e n t i t y such t h a t for  any c e l l s Then If  e  a  e , c e a  fl h(ep)  and  a  i sa  e" fl h ( i p ) = 0  e^, c ep , Z  Q  cycle i n M a n  then  a  2  e ,  L  i s transverse t o  Me^,).  homologous t o a f l ^ B.  B'= 0.  T h i s i s from g e n e r a l i n t e r s e c t i o n t h e o r y ( e . g . , see [ 1 0 ] ) .  Simple I n t e r s e c t i o n s i n G, •  and t h e P o i n c a r e D u a l i t y Map ••  K, n  A s t r a i g h t f o r w a r d t r a n s l a t i o n o f 3.2 i n t o Schubert terminology  cell  i s g i v e n below (3.4) a n d , u s i n g i t , t h e P o i n c a r e d u a l i t y map i s  d e s c r i b e d i n terms o f Schubert symbols (3.7) and some examples o f e x p l i c i t intersections are given.  3.3  Remark and N o t a t i o n :  From t h e c h a i n complex  the Schubert c e l l d e c o m p o s i t i o n C (G r  k  n  ; Z ) 2  C (G r  k } n  and a  mod 2  c o c h a i n complex  ; Z ) = Hom(C (G 2  (1.22) we o b t a i n a  r  k 5 n  ; Z ), Z ). 2  2  C^G^ ) n  mod 2  associated with c h a i n complex  For  e  0  € C (G r  k  2  element d u a l t o i.e.,  e  a * € Hom(C ( G  r  {e  k > n  ;  :  |CT| = r }  2  i s the l i n e a r  2  Q  to  1  and  e^  to  0  f o r r\ ± o.  6  i s zero  Z ) = C (G r  2  {a* : \a\ = r }  CT  ; Z ), Z )  k n  e  S i n c e t h e c o c h a i n map H (G  symbol, w r i t e t h e c o c h a i n  as 0 " & j  CT  sending  and  CT a Schubert  ; Z ),  n  k j n  mod  ;  2,  Z) 2  i s a basis f o r H (G r  for H (G r  From here o n ,  k j n  Z  map  k  n  ; Z )  dual t o the basis  2  ; Z ). 2  homology and cohomology w i l l be assumed u n l e s s  2  otherwise s t a t e d . 3.4  Theorem:  Schubert  Let  symbols.  and  e^  be c e l l s i n  G  k  for a  n  and r\  Suppose t h e r e i s an o r t h o g o n a l l i n e a r t r a n s f o r m a t i o n  £ „k+n „k+n * : R -»• R i)  e^  „ inducing  F o r any  e  $ : c  CTl  e  G  G k  n  s k  and  a  u  c  , that  n  n  , c e^ ,  e  i s transverse to  a t  #(e ). v  ii)  e  CT  n  =  * ( e ^ )  orthogonal transformations d i s t i n c t Schubert  e  C T ( 1 )  pf:  + e  H (G A  k  ) , e  n  Q  + . . . + e  C T ( 2 )  0^ e^  a ( m )  ^i  e C T  (i)  i  s  ,  m  U .  .  . U * e m  where  C T ( m )  f o r some  c r ( l ) . . . o(m)  are  \o\ + |t]| - k n . (the Lefschetz i n t e r s e c t i o n  product)  .  T h i s f o l l o w s from 3.2 as  i d e n t i t y , and  $3^7(2)  3^, . . . , §  symbols o f rank  Then i n is  U  ^ e ^ D  *  :  G  G k  homologous t o  n  e  k n  cr(i)  ^  s  homotopic t o t h e  (since  ^  i s also  homotopic t o t h e i d e n t i t y map).  3.5  Notation:  Define  S  P  : R  q  R, Q  p 5 q  by  " ^ ( e ^ =Jp - i + l f o r i 2 p e  e^  f o r i > p.  40  If 3.6  q - k + n, then Lemma:  Let e  as d e f i n e d above. i)  .  pf:  and e^ e„  CT  i f r)^ = n -  n  and <£> = §  k+n  CT _^ k  V i  +1  - n  T l ^ . i + i  then  e  CT  Vi .  fl $  = the p o i n t  {P } CT  cr and r\ a r e Schubert symbols such t h a t f o r some i .  D <£> e^ ,  CT  G ,, „ G^  ( R e c a l l 1.6.)  i ) Suppose  P € e  be Schubert De bcnubert cc ee ll ll ss ii nn  Then we have  CT^ + "H^.i+l - n - 1 If  i s t h e i n d u c e d homeomorphism.  CT  iii) G,  -+ Gy. _  n  e„ i s t r a n s v e r s e t o i>e CT T] e f l e^ = 0 u n l e s s CT^ +  ii)  in  : G^  then  dimension o f P fl ] ( R ^  )= i  ( j as i n 1.4) and  - "Hi, - + - i + dimension o f P D l ( R ~ ) = k - i + 1 k  cr.+i 1  -  CT  +  i  Tj, . +k-i+l  „  + 1  ) n K R  =» J ( R  i  +  T i  k  _  i  k  +  1  1  +  +  ) > i  1  k-  i  +  l > n  which i s a c o n t r a d i c t i o n . ii) CT^ +  T ) -  k  _  i  +  1  >  1  1x + +11  K  +  k  Thus  or  e  CT  c^  +  >  the  Since  -  1  =0.  fl  Suppose CT and rj a r e Schubert symbols such t h a t n  for  a l l  i  .  C o n s i d e r t h e graph c o o r d i n a t e s c e n t r e d a t P 1.19).  n  e^ c u C T , t h e i n t e r s e c t i o n  e  CT  H^e^  domain o f t h e graph c o o r d i n a t e s . R e c a l l t h a t  subspace c o r r e s p o n d i n g t o e . 'Claim: U D# e^ = ( L _ f i f e - ) x L  L  CT  (see 1.17,  1.18 and  lies entirely i n CT  c R  k x n  £  s  U  CT  ^he l i n e a r  CT  CT  a  c R  k x n  i n the above graph c o o r d i n a t e s .  ,  pf:  L  CT  = {A = ( a i j )  s.t.  = 0  f o r j > o^}  L  Q  = {A = (a£j)  s.t.  = 0  f o r j 5 o^}.  Suppose  A € # e^  R  c  .  The p l a n e  of t h e m a t r i x i n 1.18 c o r r e s p o n d i n g t o  A.  cp (A) = P  i s t h e row space  CT  Since  P  i s i n # e^ , i t  must s a t i s f y t h e Schubert c o n d i t i o n s dimension o f  p'n  dimension o f  Pf| l(R  i ( R ^ ) = i  and  ~ ) = i - 1.  T l i + : L  1  L o o k i n g a t t h e m a t r i x i n 1.18, i t can be seen t h a t t h e s e c o n d i t i o n s a r e independent o f  a^j  f o r j > CT^ s i n c e  f o r some  CT  a  ij  { ij  =  a  f  °  0 and  r  j  CT  j  > CT^  fl * e  . T)  $ e^ n U  CT  = CL  CT  fl $ e ) x L CT n  This c l a i m proves 3 . 6 ( i i ) s i n c e |TTJ | - (kn - |cr| ) L  Q  Vi+l  =  L  CT  fl $ e^  has d i m e n s i on  and t h e i n t e r s e c t i o n i s t r a n s v e r s e ( s i n c e  and t h u s t o  iii) T  n  i  5  for  A' € L  Thus  +  °i - -  A = ( a ^ J s . t . f o r A' = ( a l ^ ) where  CT  i  +  <  CT  « P = <P (A)  a  )] _i+i  P € * e^ fl U,  Thus  to  r  L  CT  i s orthogonal  fl <£> e ^ ) .  L  Q  Suppose CT and  r\  a r e Schubert symbols such t h a t  • -  n  V  i  P € $ e^ fl e"  CT  o P 0 j(R i  +  1  )  - 'v_i4.i P D l(R r  « P 3<e" .> rr  u  i  i n dimension + k  "  i + 1  fora l l  > i  and  n+k-o---i+l = R ) i  . . . i n dimension  k - l  for a l l  I  42 Remark:  A l t h o u g h 3.6 above shows t h a t we can always make two c e l l s  and  in  e^  G  k  transverse  n  —  $  .  However, a l t h o u g h  be a c y c l e homologous i n H.,.(G )  to  k n  union of orthogonal 3.7  Theorem:  In  i)  transformations G  k  ,  n  ii)  L  n  x H  p  cr  -* H  r  Q  ^ Z  ("1^.  fl * e^  must  0  satisfies:  and r\, - n  V i .  i s t h e map  2  k  i+1  V i  otherwise.  L  The P o i n c a r e d u a l i t y ( i n v e r s e ) map D : H -»• H n-r  P  is  CT -> TI*  pf:  CT  i t i s not i n general a  i f • T]I = n - o- _  (CT, TI) •+ ii  iii)  ep ,  a  u n l e s s CT^ + T j ^ . i + i  L  fl : H _  e  the i n t e r s e c t i o n product  f l e_ = 0  CT  the i n t e r s e c t i o n e  —  o f Schubert c e l l s as r e q u i r e d i n 3 . 4 ( i i ) .  F o r Schubert symbols e  CT  ( i n t h e manner o f 3 . 4 ( i ) ) u s i n g t h e o r t h o g o n a l  k+n transformation  e  TI . = n - CT, . , „ l k-i+1  where  V i .  ( i ) f o l l o w s from 3 . 6 ( i ) and 3.2. ( i i ) f o l l o w s from 3 . 6 ( i i ) , 3.2 and ( i ) above s i n c e i f  | cr j + |T)| = n such t h a t  and  T)^ = n - °" _i ^ k  +  does not h o l d f o r a l l i  then  3  T I . + CT, . . < n. ^0 k-iQ+1 1  ( i i i ) i s j u s t a n o t h e r way o f s a y i n g ( i i ) . T h i s r e s u l t i s e q u i v a l e n t t o P r o p o s i t i o n , page 1072 i n [3] which was f i r s t proved i n [ 9 ] .  3.8  Remark:  f : M -> M'  The i n t e r s e c t i o n p r o d u c t i s u n n a t u r a l i n t h e sense t h a t f o r a c o n t i n u o u s ( c e l l u l a r ) map,  i n t e r s e c t i o n product.  f  A  does not p r e s e r v e t h e  However, i n cohomology, t h e i n d u c e d map  f*  does  43  p r e s e r v e cup p r o d u c t , so t h a t i n t h e Grassmannians we have t h e f o l l o w i n g : i)  For  j : G  j*  -*. G  k  : H*(G , )  , ,  k  H*(G  k n  a*  -y  k j n  n' > n ,  )  as i n  j  1 . 4 ,  i s t h e map  cr*  (as can be seen from 3 . 3 , 1 . 2 1 and 2 . 1 2 ) and j * ( c r * u r\*)  ii)  = a* u TI*. 1 : G  For 1*  -»• G ,  k > n  : H*(G , ) - H*(G k  n  CT*  where  CT'  =  ( o "  n  +  k  , _  k  +  1  ,  R  ,  Q -  n  +  k  , ^  k  +  -*•  n  )  1  as i n 1 . 4 ,  i s t h e map  (CT )* 1  . . . , cr  ,  2  k  k' > k,  n + k  t)  (as can be seen from 3 . 1 , 1 . 2 1 and 2 . 1 2 ) and l*(cr* (T) '  U  T|&)  d e f i n e d from  T)  = (cr')* U as  (T)')*  cr' i s from cr).  We a l s o have a l g e b r a i c r i g h t i n v e r s e s f o r t h e maps  j * and 1 *  defined as, (j*)"  1  k f n  ) -H*(G  k f n  .)  (CT*) -»• (J,(CT))*  i s the map (l*)"  : H*(G  1  i s t h e map  : Hft(G  k j n  ) -H*(G , k  f n  )  n  <n'  and k 5k'  cr* ->- ( l ( c r ) ) * . f  These maps a r e group homomorphisms ( a c t u a l l y monomorphisms) b u t do not i n g e n e r a l p r e s e r v e cup p r o d u c t . We now go t o some s p e c i f i c examples o f i n t e r s e c t i o n p r o d u c t which use 3 . 4 d i r e c t l y .  Checking t r a n s v e r s a l i t y i s i n g e n e r a l more c o m p l i c a t e d  t o v e r i f y t h a n i n 3 . 6 , so f o r t h e remainder o f t h e paper we w i l l assume t h e following.  44 3.9  Assumption:  Let e  and e^ be Schubert c e l l s i n G^.  Q  n  suppose t h e r e i s an o r t h o g o n a l t r a n s f o r m a t i o n <S : R e Y  a  fl #  (where  § :  e  ii) such t h a t  -»• G  i s i n d u c e d by <£>) i s a  k n  |CT| + \r\\ - k n .  o f dimension i)  n  -»• R  k + n  such t h a t  k + n  ( Z ) cycle 2  Then  i s t r a n s v e r s e t o <i> e  CT  t h e r e i s an o r t h o g o n a l t r a n s f o r m a t i o n I ' : R 1  $  , and  =  e^ and f o r any e^,  c  e  a  n  a  d  e^  R  k + n  e^ , e^.,  c  t  k + n  is  t r a n s v e r s e t o <£' e^ iii)  The c y c l e  Note: 3.10  Y  is e  fl  CT  e^ .  ( i i ) =» ( i i i )  Examples:  The same n o t a t i o n as i n (2.16) w i l l be used.  Remark about Schubert c o n d i t i o n s :  R e c a l l (1.10) t h a t f o r  CT = (CT^, . . . , CT^) a Schubert symbol, t h e Schubert c o n d i t i o n s a s s o c i a t e d with  e  Q  i n G^  n  are  d i m e n s i o n o f P fl j ( R ° i ) > i  Vi .  +1  I f CTm = n , then t h e above Schubert c o n d i t i o n i s redundant ( s i n c e every  k-plane i n R  intersects  k + n  j(R  n + i  )  i n dimension i ) and  can be l e f t o u t . i)  In G  l o o k a t 12 1*1 12.  2 2  L  I f we t a k e t h e o r t h o g o n a l t r a n s f o r m a t i o n ^ 3 . 5 ) , then  12  and # ( 1 2 ) 4  : G  2  2  &  Take  12" n # (T2~) = {P € G 3  2  2  §  : G  G 2  2  2 2  (  s . t . dim. P fl <e ,  r e c a  H 3.5):  e"> _. 1  ±  2  and dim. P fl <e" , e"> > 1} 2  which i s e a s i l y seen t o be  2  (  s e e  s a t i s f y 3 . 4 ( i ) b u t n o t 3 . 4 ( i i ) , so we must use o  a different transformation.  2  X^ U X  2  where  3  45  Since  s . t . dim. P fl <e > = 1}  X  1  = {P € G  2 2  X  2  = {P € G  2 j 2  2  s- "' 1  p  ^ 1 * e" , e" >} .  c  2  X2 = 11 and X^ = $(02~) where  taking  e^ t o e~ , by 3.9 we have In G  3  2  $  i s any o r t h o g o n a l t r a n s f o r m a t i o n  12  2  ii)  3  12 = 02 + 11  L  :G  13" (1 # (23~) = {P € G  3 -> G 3 :  2  2  s . t . dim.  4  2 > 3  and dim. X^ U X  2  X  1  = {P € G  2  X  2  - {P € G  2 3  2  P fl < e ,  63,  2  e.,> > 1}  2  s . t . dim. P fl <e , e~> > 1 ±  in G  2  3  2  P c <e^, e , e g , e^>} 2  f o r <£ as i n ( i ) above, we have  2  L  1  s . t . dim. P fl <e" > = 1 }  3  X = 12 and X ^ = $ ( 0 3 )  13 f l 23 = 12 + 03  P fl <e" , e"> > 1  where  and Since  2 >  l o o k a t 13 D 23.  Here, we use  which i s  in G 2 •  . 4  iii)  In G  3  3  look a t 133 D  133" n § (233") = {P € G  233,  3  $  %  :  3  G  3 3  2  and dim. which as i n ( i i ) above i s X^ U X  using  s . t . dim. P fl <e±, e"> > 1  4  3  L  2  where  P fl < e , e g , e^> > 1} 2  :  i+6  X-L = {P € G  3  = {P € G  3  X  2  3  3  s . t . P (1 <e"2> = 1}  s . t . dim.  P 0 <e~ , e"^ > 1  P c < , e;L  However, h e r e  X  = 123  2  and  X  = #(033)  1  and  1  e , e" , e" >} 2  3  for #  4  as above.  Thus i n G  g  3  we have 133 f l iv) Let <e^, e , e > 2  3  L  233 = 123 + 033.  In  G  3 j 3  <$: R to  ->• R  look a t  123 ( 1 233. L  be an o r t h o g o n a l t r a n s f o r m a t i o n mapping  < e , e^, e^>  and  2  123 fl #(233) = {P € G  3  3  #  t h e induced homeomorphism on  s . t . dim.  dim.  and dim.  P 0 <e ,  1  = {P € G  3  3  s . t . dim.  P fl < e , e^, e^> > 1 2  P fl <e^, e , e , e^> > 2}. 2  X^ U X  X  2  = {P € G  3  3  s . t . dim.  and dim.  X  3  = {P € G  3  3  U X3  where  f  e , e , e ^ > 2} 2  3  P (1 < e , e > > 1 1  P  s . t . dim.  and  2  3  2  P fl <e^  D  2  <e^, e , e^> > 2} 2  P fl <e , e > > 1 1  . P c <i"  Under s u i t a b l e o r t h o g o n a l t r a n s f o r m a t i o n s  2  lt  #^  3  2  P (1 <e"> = 1  and dim.  3  e > > 1,  ±  With a l i t t l e d i f f i c u l t y , t h i s can be seen t o be  X  G  . . . , _" >}.  , #  2  ,  and  #  3  ,  .  47 X  ±  = ^(023),  X  Thus v)  = * (Tl3")  2  and  2  123 D In G  = # (T22).  3  3  233 = 023 + 113 + 122  T  ^  2  X  look a t  24 fl $ 34 = {P € G  34 D  24, u s i n g  L  0  $ = * : 5  s . t . dim. P fl j ( R ) > 1 3  2 > 4  P fl <e , e , e^, e^> > 1}  and dim. which i s X U Y  i n H (G  2  3  where  X = {P € G  2  4  s . t . dim. P fl <e~ , i~ > > 1} and  Y = {P € G  2  4  s . t . dim. P fl j ( R ) > 1  2  3  and X = 3>^(14)  and  Y = 23  P c j(R )}. 5  f o r some o r t h o g o n a l t r a n s f o r m a t i o n  ^  .  34 f l 24 = 14 + 23.  Thus vi)  3  L  In G  By 3 . 7 ( i ) ,  3  look a t  3  222 D 033. L  222 f l 033 = 0 L  nonempty i n t e r s e c t i o n , u s i n g  in G  3  3  3^:  222" fl $ ( 0 3 3 ) = {P € G 5  3  3  s . t . dim. P fl <e~ > = 1 5  and This i s the c e l l  § (022"), 5  , b u t l e t us t r y t o make a  but  | 022 | = 4  P c <e^, . . . , e >}s  whereas  | 222 | + j 033 [ - 9 5  = 6 + 6 - 9 = 3 .  Thus we cannot  use 3.9, a l t h o u g h  222  and  $ (033)  (the open c e l l s ) a r e t r a n s v e r s e , h a v i n g empty i n t e r s e c t i o n . 3.11  Remark:  U s i n g 3 . 7 ( i i i ) and 3.1 we can r e w r i t e t h e above r e s u l t s as  cup p r o d u c t s i n cohomology: i)  In  H*(G  2  2  ) ,  0 1 * u 01* = 1 1 * + 02*  48 ii)  In  iii)  In  iv) v)  H*(G  H*CG  In  H*(G  In  Note:  H*(G  2  3),  2  02* u 01* = 12* + 03*  3)  3  t h e same i s t r u e  3),  3  4  12* u 01* = 112* + 22* + 13*  ) , 02* u 01* = 12* + 03*.  01* u 01* = 11* + 02*  must h o l d i n  H*(G,  „) f o r a l l ,n s i n c e t h e r e a r e no o t h e r Schubert symbols o f d i m e n s i o n 2. K  k ^ 2  and  n > 2,  C o m p l i c a t e d I n t e r s e c t i o n s and t h e G e n e r a l Formula  I t i s not always p o s s i b l e t o i n t e r s e c t Schubert c e l l s as i n so t h a t 3.4 can be u s e d — 3 . 1 4 has such e x a m p l e s — a n d  3.10  f o r t h e cases where i t  i s not p o s s i b l e , a more c o m p l i c a t e d argument, such as the one d e v e l o p e d below, i s needed.  The examples  i n 3,14  3.12  For  define  Definition:  k 5 k'  g : {subsets of  G, _} -+ { s u b s e t s o f K  as  l e a d up t o t h e main f o r m u l a i n 3.16.  X c G  G, ,  \ ,n such t h a t  ,n  K  -> {P € G , ,n K ,n f o r some P' 6 X}. v  v  P  contains a  K  k-plane  3.13  j(P')c j ( R  Claim:  If  X  cr(l) + . . . + o"(m) then CT'(I)  g(x)  k + n  )  i s a cycle i n cr(i)  where  i s a cycle i n  + . . . + <j'(m)  ^(Gfc. )  homologous t o  n  a r e Schubert symbols ^ ^ . ( k ' _ ) n ^ k ' n^ G  k  homologous t o  where  CT' ( i ) = ( o ( i ) , c r ( i ) , . . 1  0  .  , cr(i), , n , n, . . . , n) v  V  k for  X = ^ ( o T T T ) U * ( O T 2 ) ) U. 2  . . . , $  1  '  - k  i = 1, . . . , m.  T h i s w i l l not be p r o v e d , but i t s v a l i d i t y  3^,  i = 1, . . . , m,  m  .  . . U * (c7TmT) m  i s suggested by t h e case  f o r some o r t h o g o n a l t r a n s f o r m a t i o n s  Here, the c l a i m i s o b v i o u s l y t r u e .  49 3.14  Examples:  i ) Look a t  24 |"I 24  in G  L  24" fl $ ( 2 4 ) = {P € G  2  X = {P € G t |  Y = {P € 24 n $(24) Although  P D <e" > = 1}  and  23 f l to  L  Thus  23 = 13 + 22  13 + 22 X  3  j  5  2  Y = j(2"3~n $(23~)) i n  By 3.9  24 fl  G  2  ^  p r e s e r v e s homology c l a s s .  i s a cycle i n  X G  2  and ^  Y  $',  so  we o b t a i n  homologous t o  then,  24 = 13 + 22 + 04  in  H,(G  0u  ) .  In cohomology, by 3 . 7 ( i i i ) , . t h i s r e a d s as 02*  ii)  u  02»  In  = 13* + 22* + 04*. G^ ^  2334 n. 2444  look a t  2334" n $ 244¥ = {P 6 G^  L  4  s . t . dim. _ dim. dim. and dim.  This i s X U Y  where  they  i s homologous  f o r some o r t h o g o n a l t r a n s f o r m a t i o n  24 fl $ 24 = X U Y  G  i s homologous t o  3  combining t h e homology c l a s s e s d e t e r m i n e d by  13+22+04.  5  . . . , e >} = 23" 0 *(23>  l s  G  (by 3.11), so  i s $'(04)  P D < e , e^, e > > 1}.  ^ , when c o n s i d e r e d as c e l l s i n  2  23*n $ (23") i n  also, since  3  $ ( 2 3 ) do not s a t i s f y t h e t r a n s v e r s a l i t y  c o n d i t i o n s i n 3.3 as c e l l s i n G do (by 3.6).  5  and  3  s . t . P c <e 23  $ = $ :  where  s . t . dim.  2}  using  2  and dim. This i s X U Y  ,  4  P fl <e±, e" , e"> > 1  s . t . dim.  4  2  using  $ = $ : 7  P fl j(R )  > 1  3  P fl j ( R ) > 2 5  P fl j ( R ) > 3 6  P fl <e , e g , y e  5  >  - 1}.  50  X = 2334 fl # 24~W fl j ( G ) = 2333 ("I $ 2333, and 43  Y = {P € 2334 fl $ 2444  s i n c e f o r P € 2334 fl § 2444, dim.  P fl <e" , e~> > 1},  s . t . dim.  P D ] ( R ) = dim.  5  i f P fl <e~ , e~> = 0, 5  P fl j ( R ) + dim.  7  6  6  then  6  P (1 <e" , e" , e~> = 4 5  6  7  P  so t h a t  must be i n X . As i n ( i ) above, 2333 f l 2333 = 1333 + 2233 L  2333 fl # 2333  2333 fl <i> 2333  in  3  i s homologous t o  (by 3.11 and 3 . 7 ( i i i ) ) , so t h a t i n G  4 4  also  i s a c y c l e homologous t o 1333 + 2233.  Y = g(233 fl  $(133) c G  3  3) f o r g  as i n 3.12.  ( T h i s can be  e a s i l y checked.) In i n 3.3, thus  G3  233 and § 133 s a t i s f y t h e t r a n s v e r s a l i t y  233 fl * ( 1 3 3 )  (by 3 . 1 0 ( i i i ) ) . G^ ^  ,  3  i n G3 3  conditions  i s a c y c l e homologous t o 123 + 033  Y = g(233 fl * 133" c G3 3) i s a c y c l e i n  Thus, by 3.13,  homologous t o 1234 + 0334. Combining t h e homology c l a s s e s o f X and Y  we have  2334 f l 2444 = 1333 + 2233 + 0334 + 1234 i n H ( G . ) . L  t  u  In cohomology t h i s reads as 112* u 2* = 1113* + 1122* + 114* + 123*.  The f o l l o w i n g two f o r m u l a s (3.15 and 3.16) c o m p l e t e l y d e s c r i b e the cohomology r i n g s t r u c t u r e i n cocycles  3.15  ( 0 , 0, . . . , 0 , a ) *  Claim:  a s a r i n g g e n e r a t e d by t h e Schubert  over a l l i n t e g e r s  a > 1.  L e tCT= ( C T ^ , . . . , c r ) and r ) = ( 0 , 0, . . . , 0 , T ] ) be k  Schubert c y c l e s i n G i)  G  k n  For j : G  . k j n  - G  k j n  , , n' > n and  k  51 ( j *)  as i n 3.8, we have  ( J * )  - 1  ii) (l*) (l*)  ( C T *  u T]*)  For  1 :G  k  CT*  u  ^ G i  n  k  T}*.  > n  •,  k' > k  and  as i n 3.8, we have  - 1  _ 1  =  (a*  uTi*)  (CT')*  =  On 1 )*  U  where CT ' = ( 0 , 0, . . . , 0,CT., . . . , CT )  and  v  v  J  v  r\ - ( 0 , 0, . . . , 0, rj, ), v  1  ,  k' - k  ^  n + k' - 1  T h i s can be p r o v e d by g o i n g t o t h e i n t e r s e c t i o n p r o d u c t v i a D (3.7) and g e n e r a l i z i n g f o r ( i ) , t h e way i n which 3 . 1 0 ( i i ) and ( v i ) g i v e t h e same answer i n cohomology (3.11) and f o r ( i i ) , t h e way i n which 3 . 1 0 ( i i ) and ( i i i ) g i v e t h e same answer i n cohomology ( 3 . 1 1 ) . Note:  ( j * -1 )  In general i t i s not true that  and ( 1 * )-1  p r e s e r v e cup p r o d u c t — s e e 3 . 1 9 ( i v ) and ( v ) .  3.16  L e t CT = (CT^ = 0, CT > 0, CT3, .  Claim:  T) = ( 0 , 0, . . . , 0, r ] ) be Schubert k  In cr*  u  .  2  H*(G)  r|* =  .  , 0 " ) and k  symbols.  we have  Z(CT')*,  summed o v e r a l l  cr' = (cr' ,  .  .  .  ,  1  symbols o f dimension  |CT'| = |CT| + r )  i = 1, . . . , k - 1  and  Indication of proof:  F o r CT and  k  CT'),  Schubert  K  such t h a t CT^ 5 cr| 5 CK  +1  for  • < CT,' CT' cr, 5 k - k • CT  T) as above, d e f i n e CT • n, as  ZCT' f o r  CT' as above.  Claim:  cr • r\ s p l i t s i n t o two sums  CT' s . t . CT' = (a£, CT^ + 1,  o'l + 1,  Z^cr' and .  .  .  ,  Z2CT'  o» + 1)  where  Z^  i s over  52 f o r a l l CT" i n t h e sum f o r (0,  Z  2  CT  2  -  1,  0-3 -  •  1,  •  •  , CT  k  -  1)  '(0, 0,  s . t . cr' = (cr^' + 1, crjj + 1 ,  i s over CT '  pf:  CT  2  -  1,  CTg  -  .  1,  .  , cr  F o r cr' i n t h e sum f o r cr • r\  t  CT' i s i n Z  0  ,  2  or  cr' < cr 12  cr^ 5 CT.! 5 o~^ ^ +  H (G. ) & k ,n  T ]  )  K  ,  = cr  2  cr' i s i n  0 , T K  1).  -  i n which  case  . 1  |CT'|  and  By 3.15, t h e cup p r o d u c t w i t h o u t l o s i n g any terms.  0,  either  i n which case  C o n v e r s e l y , f o r cr' i n Z^ satisfies  • (0,  1)  -  0,  , CT" + 1) k  f o r a l l CT" i n t h e sum f o r (0,  ,  or  Z  ,  2  i t i s easy t o see t h a t  CT'  = |CT| + r | . k  cr* u r\* can be t a k e n i n H*(G  k  n  )  Go t o t h e c o r r e s p o n d i n g i n t e r s e c t i o n p r o d u c t i n  v i a the Poincare d u a l i t y J  (3.7(iii)).  From here we can g e n e r a l i z e t h e method and r e s u l t i n 3.14(11), and we g e t  and  X  i s homologous t o t h e d u a l ( 3 . 7 ( i i i ) ) o f  Z*  Y  i s homologous t o t h e d u a l ( 3 . 7 ( i i i ) ) o f  Z* .  In t h i s way, 3.16 can be proved by i n d u c t i o n on Note:  The above f o r m u l a i n H*(G)  |CT|  h o l d s i n H*(G, K  and  r\ , .  ( C ) ; Z)  where  5TI  i t i s known as P i e r i ' s f o r m u l a (see [ 3 ] ) .  3.17  Claim: (CTCT 1  1'  In  H*(G)  we have,  cr 1*  2' • * • ' ° k  ;  CT(CT + 1 ) :  CT(CT )* k  k  -1 - D * °(Vl  CT(CT,  )ft  CT(CT - k + 1 ) * CT(CT - k + 2 ) * 1  . cr(cr + k - 1 ) * k  * °^ k-l CT  +  k  CT(CT )* 1  -2  ) "  53  the cr(a)  d e t e r m i n a n t , where t h e p r o d u c t i s cup p r o d u c t , and = ( 0 , 0, . . . , 0, a)  Indication of proof: the  for  a > 0  1  O(CT ) ' 5  2  for  a <  0.  -  1)* a ( a ) *  1  1  U CTCCT-,  1,  -  a  = (o , ±  2  2  1  2  (CT 1  o-(o\. - 1 ) *  +  )*  + (CT - 1,CT + 1)  o")  (cr-p  we have  2  CT(CT CT(CT9)"  k = 2  CT(CT + I) -  2  +  0  T h i s f o l l o w s a l g e b r a i c a l l y from 3.16 by i n d u c t i o n on  s i z e o f t h e m a t r i x ; e.g., f o r  (CT , CT )* =  and  CT2  1)  +  +  (CT 1  -  u  CT(CT  0  +  1)*  + . . . + (o, 2, CT + 2) + . 2  CT + CT ) 2  1  . + (0, o  +  2  CT ) 1  )  (The second l i n e i s from 3.16.) Note: (see  3.18  In  H  *(  G k  n  ( C ) , Z)  t h i s i s c a l l e d the determinantal formula  [3]).  Remark:  i)  The r e s u l t s i n 3.16  and 3.17  are a l s o v a l i d i n  H*(G  K  R  )  i f we use t h e p r o j e c t i o n s 1* : H*(G) •+ H * ( G )  and  K  j*  ii) CT,  i f we  : H*(G ) - * - H * ( G k  (See  They a r e a l s o v a l i d f o r  : H*(G,K, n ) -»- H*(f2(a))  i : S2(CT) -* G,  K ,n  for  ).  3.8.)  H*(S2(cr))  f o r any Schubert  symbol  use i*  3.19  k>n  Examples u s i n g 3.18  where  i s the embedding.  above:  convenience we w i l l drop t h e  The s h o r t e n e d n o t a t i o n i s used a g a i n , and 5  ' 's, c  as e v e r y t h i n g i s i n cohomology.  54  ii)  H*('S(1, 3 ) ) cohomology p r o d u c t s : 1  1  11  2  11 + 2  11  12  2  12 + 3  12  13  3  13  13  iii)  H * ( l , 1, 3 ) ) cohomology p r o d u c t s :  1 1  11  2  11 + 2  11  111 + 12  0  2  12 + 3  112  112  0  113  112 + 13  113  113  13  113  0  12 3 112  113  13  113  112 + 13  13  113 |  The next two examples a r e o f i n d i v i d u a l cup p r o d u c t s i n d i f f e r e n t Grassmannians. iv) In  124 H*(G)  u  2: i t is  1224 + 1134 + 1125 + 234 + 225 + 144 + 135 + 126.  In  H*(G  3 4  )  i t is  234 + 144.  In  H*(G  3  )  i t is  234 + 225 + 144 + 135.  In  H*CG ) j :G  (j '0  ^ ->• G  5  3  and 1 : G  ^ ->• G^  3  4  do n o t p r e s e r v e cup p r o d u c t .  5  v)  12 u 113 = (1 * 2 + 3) u 113:  In  H*(G) i t i s  + 11114  + 1223 + 1133 + 1115 + 233 + 224 + 134 + 125.  j :G  3 > 3  - G g ^ and 1 : G  —1  '1 (1*)  and Remark:  3  and ( l ' 0 ~  5  For  i t i s 1224 + 1134 + 234 + 144;  4 u  i.e. , f o r  11123  5  In G  3 j 3  -  G ^  do n o t p r e s e r v e cup p r o d u c t .  , 12 u 113 can be determined  3 3  t o be  233 (as above) by  g o i n g t o t h e P o i n c a r e d u a l s and u s i n g the i n t e r s e c t i o n method as i n 3.10. 3.20  Conclusion:  The p r o d u c t s t r u c t u r e i n H*(G), H*(G ) k  and H*(G  k  n  )  i s w e l l known from c h a r a c t e r i s t i c c l a s s e s (see [ 1 ] ) . H*(G )  i s g e n e r a t e d by t h e S-W c l a s s e s  k  OJ^, . .  . ,  of the  t a u t o l o g i c a l b u n d l e , and H"(G  k  n  )  i s generated by co^, . . . , co^.  and co^, . . . ,  under t h e c o n d i t i o n s (1 + co^ + . . . + co )(l + co + . . . + ccijj) = 1, where t h e Sj 1  k  are t h e S-W c l a s s e s o f the normal b u n d l e . I t i s known (see [2]) t h a t  COJ i s t h e cohomology c l a s s  cr(j)*  (from 3.17). By t h e map C0j  :G  k n  ->• G  which i n cohomology must map  n k  , we can f i n d t h e Schubert c o c y c l e c o r r e s p o n d i n g t o co^: (cr( j ) ) = ( 1 , 1, . . . , 1) which we can c a l l v  V  j  times  J  """(j).  «j t o  56  Thus  = co. ,  and  T(.J)*  must generate  H*(G ). k  (This  c o u l d be checked a l g e b r a i c a l l y u s i n g 3.16 and 3.17.) I t can be determined a l g e b r a i c a l l y from 3.16 and 3.17 t h a t i n ( j ) ' i s obtained r e c u r s i v e l y T(j)*  =  CT(j)*  from  Cf(j)  + CT(j - 1)*  + o(j  G,  by  u T(l)* +  - 2)* w T(2)* +  . . . +  CT(1)*T(J  -  1)*.  This r e f l e c t s the i d e n t i t y (cOj  + C0j_^ +  in characteristic  . . . + CO )(COJ+ S j _ 1  has a s i m p l e d e s c r i p t i o n . the p r o d u c t s t r u c t u r e  (iii).  +  . . . + oo^) =  1  classes.  The above shows t h a t f o r G,  description  :L  G  k  and  G^  n  ,  t h e cohomology r i n g  However, i n t h e cohomology o f Schubert v a r i e t i e s ,  i s more c o m p l i c a t e d , and t h e s i m p l e s t method o f  seems t o be t o g i v e a t a b l e f o r cup p r o d u c t as i n 3 . 1 9 ( i i )  and  57  References [1]  J . W. M i l n o r and J . D ,  Stasheff  Characteristic Classes, A n n a l s o f Mathematics S t u d i e s Princeton University Press. 12]  S. L. K l e i m a n Geometry o f Grassmannians and a p p l i c a t i o n s . . . , P u b l . Math. I . H. E. S. No, 36, P a r i s (1969).  [3]  S. L. K l e i m a n and D. Laksov Schubert C a l c u l u s , American Math M o n t h l y , 79, pages 1061-1082  [4]  (1972),  W. V. D. Hodge and D. Pedoe Methods o f a l g e b r a i c geometry v o l . I and I I , Cambridge U n i v e r s i t y P r e s s , 1953.  [5]  J . T. Schwartz D i f f e r e n t i a l Geometry and T o p o l o g y , Gordon and B r e a c h  [6]  S. S. Chern and Y u h - l i n J o u On t h e o r i e n t a b i l i t y o f d i f f e r e n t i a b l e  manifolds,  S c i . Rep. Nat. T s i n g Hua U n i v . 5, pages 13-17 (1948) [7]  S. I . A l ' b e r Homologies o f homogeneous s p a c e s , D o k l . Akad. Nauk. USSR (N.S.) 98, pages 325-328, 1954 ( R u s s i a n )  58  [8]  H. Iwamoto On i n t e g r a l i n v a r i a n t s and B e t t i numbers o f symmetric Riemannian manifolds.  I  J . Math. Soc. Japan 1, pages 91-110 (1949) [9]  C.  Ehresmann Sur l a t o p o l o g i e de c e r t a i n espaces homogenes, Ann. Math., 35 (1934)  [10]  S. L e f s c h e t z Topology American Math. S o c i e t y C o l l o q u i u m P u b l i c a t i o n s , Volume X I I New York (1930)  

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