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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Title | Some computations of the homology of real grassmannian manifolds |
Creator |
Jungkind, Stefan Jörg |
Date Issued | 1979 |
Description | When computing the homology of Grassmannian manifolds, the first step is usually to look at the Schubert cell decomposition, and the chain complex associated with it. In the complex case and the real unoriented case with Z₂ coefficients the additive structure is obtained immediately (i.e., generated by the homology classes represented by the Schubert cells) because the boundary map is trivial. In the real unoriented case (with Z₂ coefficients) and the real oriented case, finding the additive structure is more complicated since the boundary map is nontrivial. In this paper, this boundary map is computed by cell orientation comparisons, using graph coordinates where the cells are linear, to simplify the comparisons. The integral homology groups for some low dimensional oriented and unoriented Grassmannians are determined directly from the chain complex (with the boundary map as computed). The integral cohomology ring structure for complex Grassmannians has been completely determined mainly using Schubert cell intersections (what is known as Schubert Calculus).. In this paper, a method using Schubert cell intersections to describe the Z₂ cohomology ring structure of the real Grassmannians is sketched. The results are identical to those for the complex Grassmannians (with coefficients), but the notation used for the cohomology generators is not the usual one. It indicates that the products are to a certain degree independent of the Grassmannian. |
Subject |
Grassmann manifolds |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079494 |
URI | http://hdl.handle.net/2429/21381 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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