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