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9i< SOME COMPUTATIONS OF THE HOMOLOGY OF REAL GRASSMANNIAN MANIFOLDS by STEFAN JORG JUNGKIND B . S c , The U n i v e r s i t y o f A l b e r t a , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Mathematics) We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1979 © Stefan Jorg Jungkind, 1979 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or 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 gain s h a l l not be allowed without my w r i t t e n permission. Department n f Hg^it^aj-'ics The U n i v e r s i t y of B r i t i s h Columbia 207.5 Wesbrook Place Vancouver, Canada V6T 1W5 Date : - 6 B P 7 5 - 5 1 1 E A b s t r a c t When computing the homology of Grassmannian manifolds, the f i r s t step i s u s u a l l y t o look a t the Schubert c e l l decomposition, and the ch a i n complex a s s o c i a t e d w i t h i t . In the complex case and the r e a l unoriented case with Z^ 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 . , generated by the homology c l a s s e s represented by the Schubert c e l l s ) because the boundary map i s t r i v i a l . In the r e a l unoriented case ( w i t h Z^ c o e f f i c i e n t s ) and the r e a l o r i e n t e d case, f i n d i n g the a d d i t i v e s t r u c t u r e i s more complicated s i n c e the boundary map i s n o n t r i v i a l . 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 comparisons, us i n g graph coordinates where the c e l l s are l i n e a r , to s i m p l i f y the comparisons. The i n t e g r a l homology groups f o r some low dimensional o r i e n t e d and unoriented Grassmannians are determined d i r e c t l y from the chain complex ( w i t h the boundary map as 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 completely determined mainly u s i n g Schubert c e l l i n t e r s e c t i o n s (what i s known as Schubert Calculus).. In t h i s paper, a method u s i n g Schubert c e l l i n t e r s e c t i o n s to describe the Z^ cohomology r i n g s t r u c t u r e of the r e a l Grassmannians i s sketched. The r e s u l t s are i d e n t i c a l to those f o r the complex Grassmannians ( w i t h 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 the cohomology generators i s not the usu a l one. I t i n d i c a t e s t h a t the products are t o a c e r t a i n degree independent of the Grassmannian. i i i TABLE OF CONTENTS Page Abstract i i Table o f Contents i i i L i s t o f Tables i v L i s t of Figures v Acknowledgement v i In t r o d u c t i o n 1 PART I - DEFINITIONS AND NOTATION 3 Grassmannian Manifolds and Mappings between Them 3 Schubert C e l l s and Schubert V a r i e t i e s . . . 6 Graph Coordinates and Chain Complexes f o r the Grassmannians . . 10 PART I I - ADDITIVE HOMOLOGY STRUCTURE 15 General Theory f o r C e l l Complexes 15 Determining the Incidence Number f o r the Boundary Map f o r C(G, ) 17 l k, n Some Low Dimensional Examples 25 Tables of Homology Groups o f the Grassmannians 33 PART I I I - HOMOLOGY AND COHOMOLOGY PRODUCTS 37 The General I n t e r s e c t i o n Theory To Be Used 37 Simple I n t e r s e c t i o n s i n G, and the Poincare D u a l i t y Map . . 38 K 5 n Complicated I n t e r s e c t i o n s and the General Formula 48 References 57 i v L i s t o f Tables I . The Boundary Map i n C(G) . . 27 I V I I . Homology Groups f o r G . 34 I I I . Low Dimensional Homology Groups f o r G 35 IV. Homology f o r the unoriented Grassmannians 36 V L i s t of Figures 1. ( R e f e r r i n g to Example 2.4.) 17 2. ( R e f e r r i n g t o the proof of P r o p o s i t i o n 2.8.) 21 v i Acknowledgement I wish to express thanks to my supervisor Professor Mark Goresky for his help with the material presented, and also to Professors Kee Y. Lam and Jim C a r r e l l for t h e i r additional aid i n research. This paper was written while on a scholarship from the National Research Council of Canada. 1 I n t r o d u c t i o n A l o t i s known about the homology of the Grassmannian manifolds 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 Classes (see [1]) or us i n g a l g e b r a i c geometric methods (see [3] and [4]) and u s u a l l y the Schubert c e l l decomposition i s used. However, there do not seem to be r e a d i l y a v a i l a b l e answers to such questions as: a) Given a f i n i t e dimensional o r i e n t e d or unoriented r e a l Grassmannian, what i s the r - t h homology group? b) Given two cocycles i n such a Grassmannian, what i s t h e i r cup product? This paper i s concerned w i t h developing computational methods, using the geometry of the Schubert c e l l decomposition, by which e x p l i c i t answers to the above can be determined. In Part 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 chain complexes a r i s i n g from the Schubert c e l l decomposition of the U n i v e r s a l o r i e n t e d ( r e a l ) Grassmannian and the U n i v e r s a l unoriented Grassmannian. (The main po i n t i s t o compute the boundary maps.) From these complexes, the i n t e g r a l homology groups of some of the f i n i t e Grassmannians and, i n low dimensions, f o r the i n f i n i t e Gras smannians are c a l c u l a t e d . T h e o r e t i c a l l y , i t should be p o s s i b l e t o determine a l l the homology groups f o r a l l the r e a l Grassmannians ( o r i e n t e d and unoriented) from the formulas given f o r the boundary maps, but the amount of c a l c u l a t i o n r e q u i r e d increases r a p i d l y i n the higher dimensions (above 6 f o r i n s t a n c e ) . However, by l o o k i n g at the lower dimensions, i t may be p o s s i b l e to detect p a t t e r n s and make conjectures which could be proved by other 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 the 2 other 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 obtained 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 the c h a r a c t e r i s t i c c l a s s e s , e.g., which Schubert c e l l s correspond t o a given c h a r a c t e r i s t i c c l a s s . 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 determined from the chain complexes a l s o , and some examples are given. Question (b) i s completely solved i n i n t e g r a l homology f o r the complex Grassmannians i n [3] (pages 1072-1073), and the ZQ-cohomology r i n g f o r the i n f i n i t e unoriented Grassmannians i s known ([1] page 83 and [5] page 52) and some of the f i n i t e unoriented Grassmannians ([5] page 51). In Part I I I i t i s i n d i c a t e d , using i n t e r s e c t i o n products, t h a t the formulas i n [3] f o r the complex case are v a l i d a l s o i n Z^ cohomology f o r the r e a l unoriented case. The Z^ cohomology products i n the unoriented Schubert v a r i e t i e s can be determined a l s o , u s i n g the induced map i n cohomology of t h e i r embeddings i n Grassmannians. The i n t e r s e c t i o n methods used are a f i r s t step i n f i n d i n g products i n i n t e g r a l cohomology of o r i e n t e d and unoriented Grassmannians, but t h i s i s a much more complicated problem (mainly because of signs) and w i l l not be looked at i n t h i s paper. 3 PART I - DEFINITIONS AND NOTATION Grassmannian Manifolds and Mappings Between Them 1.1 D e f i n i t i o n : i ) The r e a l unoriented f i n i t e Grassmannian G, i s the k, n set of k dimensional planes through the o r i g i n ( c a l l them k-planes) i n k+n R , with topology given as follows: Let V be the set of ordered k-tuples of l i n e a r l y independent vectors i n R^ + n. V, i s an open subset of j ^ x ( n + k ) a n j - t n u s i n h e r i t s k, n i t s topology. Define an equivalence r e l a t i o n ~ on V, as A ~ B K , n (where A and B are k x (k + n) matrices) i f there i s a l i n e a r transformation of the row space of A onto i t s e l f which maps the rows of A to the rows of B ( i . e . , A ~ B i f they have the same row space). G, = the quotient of V by defines the topology f o r G i i ) The r e a l oriented f i n i t e Grassmannian G. i s the set of k, n k+n oriented k-planes (through the o r i g i n ) i n R with topology given as follows. Define an equivalence r e l a t i o n on V as above except that the l i n e a r transformation must have p o s i t i v e determinant. G = the quotient of V by *a defines the topology f o r G K,n K,n K j n k+n 1.2 Notation: i ) A representation f o r a k-plane P i n R i s a k x (k + n) matrix A having row space P. k+n i i ) A representation f o r an oriented k-plane P i n R i s as above, with the a d d i t i o n a l condition that the o r i e n t a t i o n determined by the ordered row vectors of A coincides with the o r i e n t a t i o n of P. 4 1.3 Remarks: i ) G, i s always o r i e n t a b l e , but G, i s not i n general k, n k, n (from [2] only i f k + n i s even). i i ) There i s an i n v o l u t i o n T on G, which takes an o r i e n t e d k, n plane P t o the same plane w i t h opposite o r i e n t a t i o n . Notation: C a l l T the a n t i p o d a l map, and i f Q = T(P) say t h a t Q i s a n t i p o d a l t o P. ( I n G. = S , T i s the usu a l a n t i p o d a l map.) l ,n i i i ) There i s a double covering \|r : G G, which takes an K j n K 5 ri o r i e n t e d plane P t o the same plane P w i t h o r i e n t a t i o n ignored (\|c 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 ) . Note: ( i i ) and ( i i i ) show t h a t G i s a Z bundle over G. 1.4 Mappings between the Grassmannians: The n o t a t i o n used here w i l l be used throughout the paper. i ) For i 5 p, e. w i l l denote the i - t h standard b a s i s v e c t o r I of R P. i i ) Define f : R P •+ R q f o r p - q by j ( e \ ) = , 1 5 i 5 p. For p = k + n and q = k + n' (n' > n ) , j induces embeddings j : G •+ G by j ( P ) = the k-plane j ( P ) i n R , and j : G G , by j ( o r i e n t e d plane P) = the plane j ( P ) i n J ( P ) i n R^ + n w i t h o r i e n t a t i o n induced by the o r i e n t a t i o n o f P. i i i ) Define 1 : R P -+ R q, p > q, by l ( e . ) = e l q-p+i 1 5 i < p. For p = k + n and q = k' + n (k' > k ) , 1 induces embeddings 1 : G, -*• G , n ^ G k ' , n b y 1 ( P ) = t h e k ' ~ P l a n e i n R k' + n spanned by k el» • • • > ekt_ k a n d 1(P) 5 and 1 : G, -»• G, , by 1 ( o r i e n t e d plane P) = the k'-plane i n + n spanned by e-^ , . . . , e]<'_k a n c* i ( P ) , with o r i e n t a t i o n determined by the o r i e n t a t i o n of <e~^ , . . . , !_]<•> fo l l o w e d by the o r i e n t a t i o n of 1(P) induced by the o r i e n t a t i o n of P. Note: i f k' > k and n' > n then the diagrams G k , n 1 >Gk,n' 1 1 I G k ' , n ~ — * G k ' , n and G k , n 7~* G k , n ' T commute. i v ) For each k, n there are homeomorphisms | : G]< n Gn,k t a k i n g a plane P t o i t s orthogonal complement „— . Dk+n P i n R and G\,^n -+ Gn.^ t a k i n g an o r i e n t e d plane P t o i t s orthogonal complement P"1* o r i e n t e d so that the product o r i e n t a t i o n on P x p"*~ coi n c i d e s w i t h the standard o r i e n t a t i o n on R^ + n, 1.5 D e f i n i t i o n : i ) The i n f i n i t e unoriented Grassmannian i s the union l i m i t ( v i a the embeddings j : G^ ^ n -> > n i ) as n -> °° of G^ n . i i ) The i n f i n i t e o r i e n t e d Grassmannian G^ i s the union l i m i t ( v i a the embeddings j ) as n -+ °° of G^ n . Note: These l i m i t s e x i s t since f o r n S 5 i i j , the diagrams 6 k,n j > V v S s s s ^ a n d Gk,n±—* G k , n 2 ,k,n commute. 3 k , n i ^ ^ k , n 9 By the note f o l l o w i n g 1.4 ( i i i ) above, the embeddings 1 and 1 induce embeddings 1 : G k ~" G k ' a n d 1 : G v ~* G k ' f o r k 5 k'. I t i s easy t o see that here a l s o the d i a grams and 1 V * G i < commute f o r k 5 k^ < k 2 Thus the f o l l o w i n g d e f i n i t i o n s are v a l i d : i i i ) The u n i v e r s a l unoriented Grassmannian G i s the union l i m i t as k -+ °° of G k . i v ) The u n i v e r s a l o r i e n t e d Grassmannian G i s the union l i m i t as k ->• °° of G k . \ Schubert C e l l s and Schubert V a r i e t i e s 1.6 D e f i n i t i o n : i ) A Schubert symbol CT i s a k-tuple of i n t e g e r s (o-j, . . . , CTk) such t h a t 0 5 CT1 5 . . . 5 o-k . The "dimension" of CT I S I Cr| = CT1 + . . . + 0"k . i i ) Given a Schubert symbol cr such t h a t a k 5 n, define the Schubert " c e l l " eQ i n G k > n to be the set of k-planes P i n s a t i s f y i n g the 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 the Schubert condit ,k+n 1 0 n s 7 ass o c i a t e d w i t h CT): dimension of P fl 3(R°^ + 1) = i and dimension of P fl 3(R C T^" + 1 = i - 1 f o r i = 1, . . . , k. Notation: FG i s the k-plane <eCT +^, . . . , e C T which l i e s i n e C T . Remark: The v a l i d i t y of the terms "dimensions" and " c e l l " above w i l l be shown below. 1.7 Theorem: Let k > 0 and n > 0 be given. i ) For any Schubert symbol cr = (cr^, . . . , cr^ 5 n ) , the set eQ c n i s an open c e l l of dimension |cr| . i i ) The c o l l e c t i o n of a l l such e gives G v a c e l l complex s t r u c t u r e . p f : see [1] Se c t i o n 6. ( i ) i s proved i n 1.19. 1.8 P r o p o s i t i o n : For e C T a Schubert c e l l i n G^ n , ^~ (eQ) i s a p a i r of a n t i p o d a l open c e l l s i n G^ n , each homeomorphic under \Jr t o e C T . pf: In ge n e r a l , i f f : X -> Y i s a double covering and A c Y i s c o n t r a c t i b l e , then f (A) i s a p a i r of d i s j o i n t sets each homeomorphic under f t o A. The p r o p o s i t i o n then f o l l o w s from the f a c t t h a t e C T i s an open c e l l and thus c o n t r a c t i b l e , and th a t \|/ 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 Notation: Let e C T denote the h a l f of \|r (e C T) c o n t a i n i n g the plane P C T w i t h o r i e n t a t i o n <^<j^+±> < • < » ^ a^+k-* a n c^ e o ~ denote the other h a l f ( T ( e C T + ) ) . 1.9 C o r o l l a r y : Let k > 0 and n > 0 be given. The c o l l e c t i o n of open c e l l s e C T + and e C T where CT runs over a l l Schubert symbols of the form (cr-p • • • , 5 n) gives n a c e l l complex s t r u c t u r e . p f : By 1.8, the c e l l s t r u c t u r e f o r G^ n given i n 1.7 p u l l s back v i a \|r t o a c e l l s t r u c t u r e f o r G^ n made up of the c e l l s e ( J , e C T~ f o r a l l appropriate Schubert symbols. 1.10 Claim: With the above c e l l s t r u c t u r e s f o r G^ n and G k j n , the maps _ L , j , 1, JL, j , 1, T and ^ are c e l l u l a r . Their a c t i o n s on c e l l s are _ l i e C T ) = e C T, where o^' = the number of j s . t . O j 2: i j ( e C T ) = e C T considered as a c e l l i n G^ n i l ( e C T ) = e C T i where o' = ( 0 , 0, . . . , 0, , a^, . . . , o^) k' - T j ( e C T + ) = e C T + , i ( e a + ) = e£, j ( e a " ) = e a " and l ( e a " ) = e a t > f o r a' as above T(e C T +) = e C T", i ( e C T + ) = ^ ( e ^ ) e CT J L ( e C T + ) = e j , or ea, from above, depending on cr. The statements about j , j , 1, 1, T and are e a s i l y v e r i f i e d from the d e f i n i t i o n s . The statements about J_ and _L are not so easy t o v e r i f y — o n e way i s to go t o graph c o o r d i n a t e s — b u t s i n c e they are not used i n any important way, they w i l l not be proved here. 1.11 Remark: By 1.10, the c e l l s t r u c t u r e s f o r Gj,,^ and n y i e l d i n the union l i m i t CW-complex s t r u c t u r e s f o r G^ and G^ which i n t u r n y i e l d CW-complex s t r u c t u r e s f o r G and G. (The CW p r o p e r t i e s are easy t o check, e.g., t i l page 79.) 1.12 D e f i n i t i o n : Let CT = (cr^, . . . , cr^ - n ) , a Schubert symbol, be given. i ) The unoriented Schubert v a r i e t y 2(a) i n n i s the clos u r e of e C T i n G^ n . i i ) The o r i e n t e d Schubert v a r i e t y 2(a) i n G^ n i s ^_1(°,(cr)) = the c l o s u r e of e C T + U e C T~ i n G^ n . 1.13 Remark: i ) 2(a) i n G k i s the set of k-planes P i n R k + n s a t i s f y i n g the c o n d i t i o n s dimension of P fl j ( R 1 ) - i i = 1, . . . , k and 2(a) i n G^ n i s the set of o r i e n t e d k-planes i n R s a t i s f y i n g the same c o n d i t i o n s . i i ) i f n' > n then B(n, n, . . . , n) i n G^ n t = j C G ^ ^ ) * M rv rv rv and °.(n, n, . . . , n) i n n ' = j ( G ] < ) n ) . i i i ) i f k' 2: k then °.(0, 0 0 , n, n, n) i n k times G k i j n i s l ( G k n ) and 8 ( 0 , . . . , 0, n, n, . . . , n) i n G^, n i s ~ ~ k times K G k , n ) -i v ) Suppose k' > k and n' ^  n. then j : G\,^n Gy-^i induces a homeomorphism between 2(a) i n n and S2(CT) i n G^ n i and 1 : G^ n ->• G^i n induces a homeomorphism between &(o~) i n G^ n and Q(o') i n G^t n where cr' = (v0, 0, . . . , (3, o1, a^, . . . , o^). k T - k S i m i l a r l y i n the o r i e n t e d case. 1.14 Claim: Let CT = ( c ^ , . . . , cr k 5 n ) and CT' = ( c ^ ' , . . . , CT^' 5 n) be given. Then 2(a) c S(CT') i n G v _ «=» CT- 5 CT. ' V i and 2(a) c 2(a') i n G v „ » CT- 5 CT-' V i . p f : Using 1.13 ( i ) : Suppose CT and cr' are Schubert symbols as above and CTi - °i' V i . Then P € 2(a) CT.+i => dimension of P fl j ( R ) > i V i 10 cr!+i ^ a.+i ^ a!+i => dimension of P fl ](R 1 ) > i V i , s i n c e j ( R 1 ) c J(R 1 ). Thus P € 2 ( a ' ) , i . e . , f o r a i 5 a i ' V i , P € 2(a) =» P € ffi(af) or 2(a) c 2(a<). Suppose a and a' are Schubert symbols as i n the c l a i m , and CT^ > <J2_^1 f o r some i ^ . Then P € e^ c 2(a) „ a,«rt+in-l =» dimension of P fl j ( R i 0 u ) = i _ - 1 => dimension of P D j ( R 0 U ) 5 i Q - 1, s i n c e j ( R 0 ) c j ( R 1 0 u ) . Thus P i 2 ( a ' ) , i . e . , P € eQ c 2(a) =» P ? 2 ( a ' ) , or 2(a) <Z 2(a»). The same arguments work f o r . t h e o r i e n t e d case. 1.15 Remark: We can consider a l l the Schubert v a r i e t i e s as f i n i t e dimensional subcomplexes o f the u n i v e r s a l complexes G or G, since the i n c l u s i o n maps j , j , 1 and 1 are homeomorphisms on any Schubert v a r i e t y . The i n c l u s i o n s between the 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 the Hasse diagram. The diagram i s v a l i d f o r both o r i e n t e d and unoriented 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^ ^ up to dimension 8. Note the 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 the map J_ c e l l w i s e There i s a l s o a v e r t i c a l symmetry—the other h a l f of the diagram f o r Schubert v a r i e t i e s i n G^ ^ can be obtained by r e f l e c t i n g across dimension 8. This comes from Poincare D u a l i t y . Graph Coordinates and Chain Complexes f o r the Grassmannians 1.17 D e f i n i t i o n : Graph coordinates f o r G^ n . F i x the standard b a s i s on R k' n. Let P be a k-plane i n R k' n. Define the graph coordinates centred at P as f o l l o w s : l e t P"*~ be the orthogonal complement of P i n R n + k 11 12 and h : P x P*- -»• R k + n the isomorphism h(v, w) = v + w (vector a d d i t i o n ) Define cpp : R k X n fa Hom(P, P1") -* G k > n by <Pp(f : P •+ P 1) = h ( g r a p h ( f ) ) , a k-plane i n R k + n i ) This g i v e s the graph coordinates centred at P f o r G k n . i i ) I f P i s an o r i e n t e d plane, define ( P p : R k x n G k j n as above, g i v i n g <pp(f : P -> P 1) the o r i e n t a t i o n induced by the o r i e n t a t i o n of P v i a the isomorphisms P & graph(f) ^ h ( g r a p h ( f ) ) . This gives the graph coordinates centred at P f o r G k j n . 1.18 Remark: In the graph coordinates centred at P a , there i s a n a t u r a l choice of isomorphism Hom(PCT, P C T ) R as f o l l o w s . Give P^ the ordered b a s i s { ea^+l' ®cr2+2» • • • » ^ o"k+k} a n d § i v e PQ- t n e ordered b a s i s of the remaining b a s i s vectors i n R k + n ( i n i n c r e a s i n g order a l s o ) . Let f : P C T ->• P C T correspond t o the matrix of f i n the above bases. Then w i t h t h i s correspondence, the map <PCT (= <Pp ) : R G k n (or G k n ) i s given by A = ( a ^ j ) -> the k-plane w i t h r e p r e s e n t a t i o n (see 1.2) a l l a12 ' ' ' a l a 1 1 a l o 1 + l • • • a i a 2 0 a l o 2 + l • • • a l o " k 0 a i c k + l • • • a l n  a21 a22 • • • a 2 o 1 0 a 2 o 2 + l • • • a 2 c r 2 1 a 2 a 2 + l • • • a 2 o k 0 a2o" k +l • • • a 2 n • • • * • 0 • • • • • l * k l a k 2 ' • ' a k a k 0 a k a k + l • • • a k a 2 0 a k o 2 + l • • • a k o k 1 ^ o ^ + l • • • a k n 13 and (P C T(R k > < n) i s the set of planes having 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 k n a l s o . ) Notation: C a l l c p a ( R k x n ) U 0 i n G k j n and U C T + i n G k ? n . Give R the ordered b a s i s { A ^ , A 1 2» • • • » A l n , A21, . . . , A k n } where A^j = matrix w i t h 1 i n i j t n p o s i t i o n and zeros everywhere e l s e . 1.19 Claim: Let o = (o^ , . . . , o"k _ n) be a Schubert symbol. Then i ) e C T c Uct i n Gk^n and cpQ (eQ) i s the plane » A-lnr. > A ? 1 , A00, . . . , A- , A„. , . . < A A 11' 12' * " * ' "l 0"^' "21' n22' ' * • » "202' "31' * * * » i n R above. kxn A k l ' • ' • ' A k o k > C a l l t h i s plane L^ . i i ) In G k > n , e C T c TJCT and cp~ ( e a + ) i s the same plane L Q pf: I t i s easy t o see th a t a plane P € G k n e C T » i t has a r e p r e s e n t a t i o n of the form i s i n 1 0 0 * 'CT 1 +1' column . 0 0 0 . * 1 0 0 * . " 0 column 0 0 0 0 1 0 0 * 0 0 . , * 1 column <=» P = <pCT(A) f o r some matrix A = ( a i j ) such that a j_ j = 0 f o r i > CTJ_ + i + 1 . This proves part ( i ) . Part ( i i ) i s proved the same way, no t i n g that P a l i e s i n e ^ . 14 1.20 D e f i n i t i o n : Let CT = (cr^, . . . , 5 n) be a Schubert symbol. Define the o r i e n t a t i o n of eQ i n G k n and e a i n G^ n t o be t h a t induced by L C T , where L C T i s given the o r i e n t a t i o n determined by the ordered b a s i s vectors which span i t . Carry over the o r i e n t a t i o n of t o e C T~ v i a T. 1.21 Remark: i ) We now have c e l l s t r u c t u r e s of o r i e n t e d c e l l s f o r G k n and G k ? n . i i ) The maps j , j , 1 and 1 ( a l s o and T) 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 these c e l l o r i e n t a t i o n s induce c e l l o r i e n t a t i o n s i n G^ , G^ , G and G. 1.22 D e f i n i t i o n : i ) Define the graded group C ( G k > n ) as C r(Gj < n ) = the f r e e a b e l i a n group generated by the Schubert c e l l s e C T i n G^ n of dimension r . Define C(G k) and C(G) s i m i l a r l y . i i ) Define the graded group C ( G k ? n ) as the f r e e a b e l i a n group generated by a l l the Schubert c e l l s e C T + and e C T~ i n n of dimension r . Define C(G k) and C(G) s i m i l a r l y . Remark: These graded groups are the b a s i s of chain complexes f o r the Grassmannians a r i s i n g from the o r i e n t e d c e l l decompositions. 15 PART I I - ADDITIVE HOMOLOGY STRUCTURE In t h i s s e c t i o n , the a d d i t i v e s t r u c t u r e o f the i n t e g r a l homology of G^ n and G^ n 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 o r i e n t a t i o n s the boundary homomorphism . d f o r the chain complex (G^ n ; Z) a r i s i n g from the Schubert c e l l decomposition. The formula f o r d (Theorem 2.9) i s the main r e s u l t aimed f o r , and then some low dimensional homology groups f o r Gj,. n and G^ . n are derived. General Theory f o r C e l l Complexes In g e n e r a l , given a CW-complex K together w i t h an o r i e n t a t i o n f o r each c e l l i n K, there i s a homomorphism d : C 1(K; Z) ^•C1~"''(K; Z) making C i n t o a chain complex so t h a t the homology of ( C , d) ® G i s H A(K, G) f o r any group G. (This can be done, f o r example, by t r i a n g u l a t i n g K so that the c l o s u r e of each c e l l i s a f i n i t e subcomplex, and using s i m p l i c i a l methods t o define d, see [6].) 2.1 D e f i n i t i o n : Let K be a CW-complex w i t h o r i e n t e d c e l l s , and d the r e s u l t i n g boundary homomorphism. For e Q and e^ c e l l s of dimension r and r - 1 r e s p e c t i v e l y , define the incidence number [e^, e R] t o be the e p - c o e f f i c i e n t of de a • a • For the o r i e n t e d Grassmannians, the only p o s s i b i l i t i e s w i l l be [ ea> e p l = 0 o r ±1> which the f o l l o w i n g f a c t s w i l l take care of: 16 2.2 Let K be a CW-complex of dimension n and e a , ep o r i e n t e d c e l l s of dimension r and r - 1 r e s p e c t i v e l y . i ) I f e Q D ep = 0 then [ e Q , ep] = 0 i i ) I f there i s an open set U i n k and a homeomorphism <p : ( R n , H +, L) •* (U, e a fl U, ep fl U) where H + i s a l i n e a r r - h a l f space and L a l i n e a r r - l space bounding H +, then r [ e a , e p ] < 1, i f the o r i e n t a t i o n of H + induced by e a c o i n c i d e s w i t h the o r i e n t a t i o n of L induced by e^ followed by the normal of L i n H -1 otherwise. i i i ) Let e a l s o be an o r i e n t e d c e l l of dimension r , and suppose there i s a homeomorphism <p : ( R n , H +, H", L) •+ (U, e a fl U, e y n U, e p fl U) where L i s a l i n e a r r - l space and H = H + U L U H~ i s a l i n e a r r space. Give H + and H- o r i e n t a t i o n s induced by e a and e r e s p e c t i v e l y , Then [ e Q , ep] = [e^, ep] i f there i s a change of o r i e n t a t i o n across L i n H • t e Y ' e p l otherwise These f a c t s w i l l not be proved here, but can be checked by going to a s i m p l i c i a l d e f i n i t i o n of d (see, f o r example, 6 ). 2.3 I f k and k' are CW-complexes with o r i e n t e d c e l l s , and f : k -+• k' i s a c e l l u l a r continuous map t a k i n g r c e l l s t o r c e l l s p r e s e r v i n g o r i e n t a t i o n s , then f induces a chain map f # : C i ( k ) •+ C i(k') V i - ( i . e . , f # o d = d ° f # ) and i f f i s s u r j e c t i v e or i n j e c t i v e then so i s f g . This i s a very weak form of the n a t u r a l i t y of the chain complex q ( k ) Determining the Incidence Number f o r the Boundary Map f o r C-(G V ) The manner i n which the general theory i s a p p l i e d t o G, i s K ,n best explained by an example. 2.4 Example: The boundary map i n C^(G 2 '• Rather than use the c e l l o r i e n t a t i o n s given i n 1.20, i t i s convenient t o define the o r i e n t a t i o n s as we proceed. Consider the graph coordinates centred at p ( Q o ) ' w h e r e ^(0 0) a l l a12 = "t^ie ( o r i e n t e d ) row space of *21 a22 1 0 a 11 a12 O l a 21 a22 I t i s . e a s y t o see t h a t the c e l l (0, 1 ) + corresponds t o the l i n e a r h a l f space < A 2 i > | a ( r e c a l l A^j from 1.18), and ( o , 1 ) ~ corresponds to <A 2 1> a2i<0 ( o , 2) + corresponds to <A 2 1, A 2 2>| (1 , D + corresponds to < A n , A 2 i > | ( 1 , D~ corresponds to < A11» A 2 1>| ( o , 2)" corresponds to <A 2 1, A 2 2 > | Fig. 1 a^-]>0 a 2 2<0 Gi v i n g these 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 the incidence numbers [ ( 1 , 1 ) + , (0, 1 ) + ] , [ ( 1 , 1 ) + , ( 0 , I ) - ] , e t c . and thus ob t a i n d ( l , 1 ) + , d ( l , 1 ) " , d(0, 2 ) + 18 and d(0, 2 ) ~ . In order t o determine d ( l , 2 ) + and d ( l , 2 ) ~ , a more complicated procedure i s needed since ( 1 , 2 ) + i s not a l i n e a r h a l f space i n these coordinates ( i t i s the set a l l a12 A : I At = 0 We must | a 2 1 a 2 2 j go to new graph coordinates where the c e l l ( 1 , 2 ) + i s l i n e a r as w e l l as (1 , 1 ) + and (0, 2 ) + ( P ^ Q ^ w i l l work), and keep t r a c k of the o r i e n t a t i o n s induced by the c e l l s ( 1 , 1 ) + , ( 1 , 1 ) ~ , (0, 2 ) + and (0, 2 ) ~ on t h e i r corresponding l i n e a r subspaces i n the new coordinates. This i s the main t e c h n i c a l i t y i n the proof of P r o p o s i t i o n 2.8. Here, we can i n a s i m i l a r manner obt a i n d ( l , 2 ) + , d ( l , 2 ) ~ , d(2, 2 ) + and d(2, 2 ) ~ which together w i t h the above w i l l y i e l d the homology of G 2 2 s t a t e d i n 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 the same as i n 2.9 as the choice of 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: For o~ = (CT^ , . . . ., 0"k) a Schubert symbol, denote by o - 6 S the symbol (CT1 - 6 l g , . . . , - 6 k g ) where 6.g i s the Kronecker 5. Note: CT - 6 S i s a Schubert symbol » CTS S CTS - 1. 2.6 Lemma: Let cr = (CT^ , . . . , S n) and cr' be Schubert symbols such t h a t |CT| = |CT'| + 1. Then i n G k n , [e*, ej,] = [ e * , e C T i ] = 0 unless cr' = cr - 5 g f o r some s. pf: This f o l l o w s from (the proof of) 1.14 and 2.2 ( i ) and the f a c t that i f cr' / cr - 6^ , f o r any s then a. = cr! + 1 f o r some i„ . s i 0 i 0 0 2.7 Lemma: Let o = (cr^, . . . , o^ 5 n) and cr' be Schubert symbols such t h a t CT' = CT - 5 s f o r some s. Then i n G k n , (wi t h c e l l o r i e n t a t i o n s as defined i n 1.20) we have 19 i ) [e+, ej,] = (-l)°s+k-s[e-, e+,] i i ) [e+, e+,] = (-1) 3 + 1 3 + 2 ^ pf: This i s proved by going t o graph coordinates where the lemma takes the f o l l o w i n g form. 2.8 P r o p o s i t i o n : Suppose CT and cr' are Schubert symbols as above and |CT'| = r . Then i n the graph coordinates centred a t P t' we have: i ) ea, c U C T i and i s a coordinate r-plane i n R^ X n. As i n 1.19 c a l l t h i s plane LQ, . i i ) e C T + fl U C T i and e ~ (1 U » are coordinate r + 1 h a l f planes ( c a l l them H + and H~) such t h a t H + U L , U H~ i s a coordinate r + 1 plane. i i i ) The o r i e n t a t i o n s of H + and H induced by e C T + and are the same eCT » o"s + k - s i s odd. i v ) The o r i e n t a t i o n of H + induced by e C T + c o i n c i d e s w i t h the o r i e n t a t i o n of L_ (as defined i n 1.20) followed by the normal i n t o H + <=» k - s + CTG+1 + CTS+2 + . . . + CT^ i s odd. Note: The above, together w i t h 2.2 ( i i ) and ( i i i ) immediately proves 2.7 pf: Using the above n o t a t i o n : i ) This i s 1.19. i i ) P € e C T + fl UCT, » i t has rep r e s e n t a t i o n s (see 1.2 and 1.18) of the forms 20 X l = row s * 1 0 0 * ft o * 0*^ +1 column 0 0 0 0 . . 1 0 0 * V s - 1 CTs+s column column * 1 0 o k + k column and row s 1 0 ft o ft crj+1 =CT1+1 column CT'+S s o'+s+l s -O+s-1 =cr„+s S column column ft 1 ft CT'+k k =a k +k column » there 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 t a k i n g the r e p r e s e n t a t i o n of P t o an X 2 r e p r e s e n t a t i o n > 0 i n the r e p r e s e n t a t i o n . (This can be seen by l o o k i n g only at the columns + i V i , and cr + s - 1.) In such a case, the X 2 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 l ' s except f o r row s which w i l l have a p o s i t i v e number (1/©) i n column CTs + s, and O's t o the r i g h t . Thus P = <P^t(A) where A = ( a ^ j ) i s a k x n matrix with 21 a,--: = 0 f o r j > a- + i + 1 and a,, ^  > 0. -L J _L t> , U o Conversely, f o r any such matrix A , P = cp^,(A) has r e p r e s e n t a t i o n s of the forms X^ and X 2 . S i m i l a r l y , P € e a ~ fl U_, » T(P) has a r e p r e s e n t a t i o n of the form X^ and P has a r e p r e s e n t a t i o n of the form X 2 «• there 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 the X^ re p r e s e n t a t i o n t o an X 2 r e p r e s e n t a t i o n ~ © < 0 » P = < P C T , ( A ) where A = ( a ^ j ) i s a k x n matrix w i t h a^j = 0 f o r i _ CT • + i + 1 and a„ _ > 0. J l S,CTS This proves ( i i ) . From ( i i ) we have the diagram shown. For ( 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 of H and H induced by e C T +, which can be done by f i n d i n g the Jacobian of the maps -1 -1 <P- ° <PCTt and cpCT o T o cp^, . ( H + and H have n a t u r a l o r i e n t a t i o n s given by t h e i r ordered b a s i s <A 1 1 ' ' A l a 1 ' A 2 l ' ' k l ' ' Aka k>' The o r i e n t a t i o n of coordiyvxV« induced by e C T + w i l l be the same i f _ i cp o cp , has Jacobian w i t h p o s i t i v e CT CT F i g . 2 determinant, and the o r i e n t a t i o n induced by e^ - w i l l be the same i f cp ° T ° <p t has Jacobian with p o s i t i v e determinant.) To w r i t e these maps O C T coordinatewise, 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 another. Given P € UCT fl U _ i , l e t v^, . . . , v k be rows of the type r e p r e s e n t a t i o n f o r P. To o b t a i n a r e p r e s e n t a t i o n of the form X 2 , 22 use rows w1, . . . , w k where the w-'s are obtained i n the f o l l o w i n g way: w r i t e the **s i n X 1 as a^. i n the appropriate manner -1 (A = (a i :-) w i l l be <Pa,(P) and a s > Q w i l l be © — s e e 1.18). Then w„ v s / a s , o s a n d w £ = vj_ - ( a i C T / a s ) ( v s ) i * s iCT s'"so- s / v "s-Note: the determinant of t h i s t ransformation i s l / a s o - , and 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 S C J > 0 and o r i e n t a t i o n r e v e r s i n g i f scr s Working out t h i s l i n e a r transformation i n c o e f f i c i e n t s , we get cp;*(P) = ( a l m ) - ( b . J = cfTV) W h e r e ~ a i o -so ser f o r i = s, j f o r i i s, j = CT f o r i = s, j f cr a . • a. s] ICT scr„ f o r i / s. n i cr •1 2.8a. For convenience c a l l t h i s map f . Then f l a > 0 i s ip" o m scr and f a < 0 i s T o c p . o c p so cr cr s We must now f i n d the determinant of the Jacobian when we r e s t r i c t f to L . For f i j ( a l m ) = b i j , 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 f o r 1 1 s, m i os , 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 gives us the i d e n t i t y 23 matrix so f o r the determinant we need only worry about the k - s + c r x k - s + cr matrix s s 3f 3a lm where i = s or j = a, , 1 = s or m = as , and j < a. and m 5 CT-^ This matrix i s 3f s i S a s l 3 a s 2 r -1 ( a _ ) af s2 af so af S+lCT af ko SO~_ (a-s> -1 3a 3a„,., _ . . . 3a, „ so\ S+1CT ko s s s 0 (a„„ ) S0\, - ( a _ ) -2 • ( a _ ) S 0 0 0 the 3a„^ column scr„ -1 2.8b. The determinant of J f | T i s thus ( - l ) k ~ s + 1 ) / ( a o r r ) | J J C T S E o\,+k-s+l 2 . 8 ( i i i ) . From 2.8b and 2.8a above we have t h a t the o r i e n t a t i o n s on H and H agree o~s + k - s i s odd. 2 . 8 ( i v ) . Comparing f i r s t the o r i e n t a t i o n <A 11 » ' * * ' A 1CT . ' A ? 1 > • * ' ' A 9 r r _ > • • • > A H > lCT  "21' - ' ' ' "2c ' • • * ' "kl  * * ' ' Ako\ > o f H  1 ^ k with the o r i e n t a t i o n < A11> • • • > Alo 1» A21> , A s l , . . . , A s C T^_ l S A g + 1 , . A k l , . . . , A k a > of L^, followed by A g o- (the normal i n t o H +) we have agreement CTs+l + CTs+2 + + Oy. i s even. Comparing the above o r i e n t a t i o n f o r 24 H +, with t h a t induced by e*, by 2.8b we have agreement » ( - l ) k - s i s odd. Combining the two, we have agreement —• k - s + o" s + 1 + CTs+2 + • • • + CTk ^ s Q-E.D. 2.9 Theorem: The boundary map d i n the chain complex f o r G k n w i t h c e l l o r i e n t a t i o n s as i n 1.20 i s ,, +x V / i xl+k-s+o c : +<i + . • .+Ov/ + / „ xk-s+o s _ d(e_) = I (-1) s+1 k ( e C T _ 5 + (-1) e a _ 5 ) s s . t . s s and d(e~) = Td(eJ) pf: This f o l l o w s d i r e c t l y from 2.7 and 2.8 and the 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 C o r o l l a r y : The boundary map d i n the chain complex f o r G k n w i t h c e l l o r i e n t a t i o n s as i n 1.20 i s v . 1+k-s+CT + 1+. . .+o-k k-s+cr„ d(e_) = I (-1) 3 + 1 K ( l + (-1) 3 ) e C T _ 5 . s s . t . s CT ,5CT -1 s-1 s pf: This f o l l o w s from 2.9 and the f a c t t h a t >|r : G k n G k n maps e* and e C T t o e C T p r e s e r v i n g o r i e n t a t i o n . 2.11 Remark: For j , 1, j and 1 the embeddings i n 1.4, the induced chain maps commute with d (from 1.21 and 2.3) so th a t the above formulas are v a l i d i n G k and § k , and a l s o i n G and G ( i f we t h i n k of 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 CT^ t o avoid the problem of having an i n f i n i t e number of CT^). 25 Some Low Dimensional Examples Fi n d i n g the homologies of the unoriented 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 computation which w i l l be c a r r i e d out over Z i n some examples below. In g e n e r a l , homology over other groups can then be determined using the U n i v e r s a l C o e f f i c i e n t s theorem, but i n the f o l l o w i n g case i t i s e a s i e r t o compute the homology d i r e c t l y from the chain complex: 2.12 Theorem: H r(G; Z 2) = C r(G; Z 2) f o r a l l r , and the same i s tru e f o r G^ . and G^^ f o r a l l k and n. pf: From 2.10, the boundary map d i s 0 mod 2 i n a l l dimensions. The method used i n the examples i s t o f i n d i n dimension r a set of f r e e generators f o r the group of c y c l e s (denoted Zp) and w r i t e out the boundaries d ( C r + ^ ) (denoted B r) i n terms of these generators. The homology H p = Z r/B r i s then the set of generators of Z r together with r e l a t i o n s given by s e t t i n g the boundary elements to zero. The main d i f f i c u l t y i s i n l o o k i n g f o r a set of f r e e generators f o r Z r , as i t i s not always c l e a r whether or not a set of cy c l e s spans the whole of Z p (although 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 the examples given i s obvious, and i t i s easy t o determine what the rank of Z r should be). In a l l the cases worked out, the above poi 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 higher dimensions i s not p o s s i b l e . 2.12a Note: In G, , when w r i t i n g boundary elements do and da (where \a\ = r + 1) i n terms of generators f o r Z r , i f da+ = ±da~ then we need only worry about da+. Thus i n w r i t i n g the boundaries i n terms 26 of generators of Z r , some Schubert symbols y i e l d two expressions and some only one. In Table I and i n the examples 2.15, a shortened 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. (as 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 confusion) and le a d i n g zeros w i l l be omitted. The zero symbol w i l l be denoted i i ) In G, the symbols +-o~ ( s i m i l a r l y -+a, ++a and —CT) w i l l r e f e r t o a l i n e a r combination of a n t i p o d a l Schubert c e l l s where the f i r s t s i g n r e f e r s t o the c o e f f i c i e n t of CT+ and the second s i g n to that of CT~. A Schubert symbol cr with one (or no) ; s i g n attached t o i t w i l l r e f e r t o the p o s i t i v e c e l l c r + ; e.g., +-23 r e f e r s to the chain element (2, 3 ) + - (2, 3)' 23 + 14 r e f e r s t o the chain element (2, 3 ) + + ( 1 , 4 ) + TABLE I: THE BOUNDARY MAP IN C(G): (NOTATION FROM 2.13, "-*» REPRESENTS d ) . 1 2 3 4 5 6 7 8 9 10 G2,2 1-*-+* 2-+--1 11-*—1 12-*—11 ++ 2 G2,3 3-*-+ 2 13-*-+ 12 — 3 23-*-+ 22 -+ 13 33-*— 23 G2,» 4-*— 3 14-*— 13 ++ 4 24-*— 23 +- 14 34-*— 33 ++ 24 44-*+- 34 G2,5 5-*-+ 4 15-*-+ 14 5 25-*-+ 24 -+ 15 35-*-+ 34 — 25 45-*-+ 44 -+ 35 55-*— 45 G3,3 112-)— 111 +- 12 113-*-+ 112 -+ 13 122-*+- 112 -+ 22 123-*-+ 122 -+ 113 +- 23 222*— 122 223-*-+ 222 + +123 133-»— 123 -+ 33 233-*— 223 — 133 333-*-+ 233 114-*— 113 +- 14 124-*— 123 +- 114 -+ 24 134-*— 133 ++ 124 +- 34 224-*— 223 — 124 144-*+- 134 -+ 44 234-*— 233 ++ 224 ++ 134 244-*+- 234 — 144 334-*— 333 +- 234 G3,5 115-*-+ 114 -+ 15 125-*-+ 124 -+ 115 +- 25 135-*-+ 134 — 125 -+ 35 225-*-+ 224 ++ 125 145-*-+ 144 -+ 135 +- 45 235-*-+ 234 — 225 — 135 1111-*--111 1112-*—1111 ++ 112 1113-*-+1112 — 113 1122-*+-1112 — 122 1114-*—1113 ++ 114 1123-*-+1122 -+1113 ++ 123 1222-*—1122 ++ 222 1124-*—1123 +-1114 — 124 1133-*—1123 . — 133 1223-*-+1222 ++1123 — 223 2222-*+-1222 1134-*—1133 ++1124 ++ 134 1233-*—1223 —1133 ++ 233 1224-*—1223 —1124 ++ 224 2223-*-+2222 -+1223 1144-*+-1134 — 144 1234-*—1233 ++1224 ++1134 — 234 1333-*-+1233 — 333 2233-*—2223 +-1233 2224-*—2223 +-1223 28 2.14 Theorem: i ) H r(G ; Z ) =4 Z f o r r = 0 and n 0 otherwise i i ) H r ( G l 9 n , Z) =L Z f o r r = 0 and r = n i f n i s odd Z 2 f o r r < n and odd 0^ otherwise. pf: i ) z r ( G l j n , Z) are generated by the chain elements ( r ) + ( r ) f o r r odd, r - n, and ( r ) - ( r ) f o r n > r > 0 and even. For r = 0 Z p i s generated by ( 0 ) + and ( 0 ) ~ . (This i s e a s i l y seen from the Table I.) The boundary group B r (Image of d : C r + 1 -> C p) i s generated by ( r ) + + ( r ) f o r r odd r < n ( r ) + - ( r ) f o r 0 5 r < n. Thus H r = Z r/B r i s zero except f o r r = 0 and n where i t i s Z. i i ) From 2.10, the f o l l o w i n g can be v e r i f i e d r Z generated by B generated by Thus Z r/B r i s ( r ) r odd i 0 r > 0 even (0) (the 0 - c e l l ) f o r r = 0 2(r) r < n odd 1° r even or r = n < Z 2 f o r r < n and odd Z f o r r = 0 and r = n i f n i s odd 0 otherwise. v . 29 2.15 Examples: In these examples, Table I i s used by i n s p e c t i o n t o f i n d generators f o r Z r . In l a b e l i n g the 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 of d i f f e r e n t dimensions (e.g., both ++1 and +-2 are l a b e l e d " a " ) . As i t w i l l always be c l e a r what dimension i s being t a l k e d about, t h i s should not cause any confusion. Notation i s as described i n 2.13. i ) 5 2 > 3 : Dimension: 0 1 2 3 4 5 6 Generators ...+ a=" a=++l a=+-2 a=++3 a=++22 a=++23 a=+-33 f o r Z p : b=*~ b=+-ll b=+-12 b=+-13 c = 2 - l l +-22 B : r a+b d(2)=-a d(3)=-a d(12)= 2c-a-b d(13) +=-b-a d(13)~=b-a d(22)=b d(23)=-b d(33)=-a I t i s easy to see that Z. r / B r = Iz r = 0, 2, 4, 0 otherwise v. (In dimension 2, Z r/B r has the r e l a t i o n s a = 0 2c c generates Z r / B r and has order 0.) Thus In the next examples only the f i r s t homology groups are determined, as the r e s t can then be found using 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 s i n c e G v _ i s o r i e n t e d f o r a l l k, n. i i } 53,3 Dimension ( r ) : 0 1 2 3 4 Z r : a=*+ a=++l a=+-2 a=++3 , a=++22 b=*~ b=+-ll c = 2 - l l b=+-12 c=++lll b=+-13 +-22 c=+-112 -+22 B r : a+b a a 2c-a-b 111: b b±a b 112:. b±c b 113: b+c 122: c See note 2.8a. r V B r Z 0 f o r r f o r r f o r r 0, 4 2 1, 3 i i i ) G 3,4 Dimension ( r ) : 3 4 5 6 Z : r a ,b,c 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 B : r 4: a, b ,c ,b+c a, 34: e b±a,b 14: 2d-b+e-a 24: a±e 124: c±d b±c 114 b±e 133: c±a 222:±c 223: c±b 123: 2d-c -bta The unexplained elements i n Z r and B p are from Examples ( i i ) and ( i above, using the same l a b e l s . r H r = Z r/B r = < 0 r = 3 Z © Z r = 4 r = 5 and 6. The homology groups H Q , H 1 and H 2 are the same as i n G, 32 i v ) SKI, 4): Dimension ( r ) : 1 2 3 4 5 C r : 1+ l l + ; 2 + 1 2 + ; 3 + 1 3 + ; 4 + 14 + and a n t i p o d a l s a=++l a=+-2 b=+-ll c = 2 - l l a=++3 b=+-12 b'=++13—4 e=+-4 e=+-14 B r : 11: a 2: a 12: 2c-a-b 3: a 13: b±a 4: a 14: b' H : r 0 Z 0 Z Z v) 2 ( 1 , 2, 3): Dimension ( r ) : 1 2 3 4 5 6 C r : 1 + l l + ; 2 + 1 2 + ; 3 + ; l l l + 13 +;22 +;112 + 23 +;122 +;113 + 123 + and ant i p o d a l s Z : r a a;b;c a;b; a=++22 a=++23 c =++123 c=++lll b=+-13+-22 b=++113 c=+-112-+22 c=++122 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 2 0 z Z©Z©Z Z 33 v i ) G^ i | : Dimension 6 Cycles: a, b , 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 + 2 4 - 3 3 g = ++1113—114 and h = ++1122—222. Boundaries: e , c ± d , c ± a , c ± b as b e f o r e , and i n a d d i t i o n , 1114: g, 1222: h 1123: 2 f - h - g - e ± c . Thus i n homology we have e = g = h = 0, a = b = c = d, 2 f = c and 2c = 0. Thus f generates H g and 4f = 0, 2f £ 0, so H g = Z^ . Tables of Homology Groups of the Grassmannians Table I I I t a b u l a t e s the above r e s u l t s together w i t h a few more tha t have been worked out i n the above manner. By going t o large enough Grassmannians, such r e s u l t s are v a l i d f o r G^ and G as shown i n Table I I I . 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 the r e s u l t s can be compared w i t h those obtained 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 (see [ l ] pages 179 and 182). The copies of Z are generated by P o n t r j a g i n c l a s s e s and t h e i r products. Another method would be t o use the cochain complex d i r e c t l y , where the incidence numbers d e f i n i n g 6 (the coboundary map) would be [ e a , ep] = [ep, e a] from the boundary map. Going through the same procedure as i n Example 2.15, e x p l i c i t generators i n terms of Schubert c e l l duals could be determined. In t h i s way f o r instance i t could be found which Schubert v a r i e t i e s represent the 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 G, where k and n are s m a l l 52,2 G2,3 G2,4 g 2 , B S,3 G3,<4 S,5 Ho Z Z z Z Z Z Z z H l 0 0 0 0 0 0 0 0 H 2 z e e z z Z Z 2 Z 2 Z 2 Z 2 H 3 0 0 0 0 0 0 0 0 H<4 z z z @ z Z z z ® z z z « z © z H 5 0 G 0 z Z 2 Z 2 . Z 2 H 6 z Z z Z 2 Z 2 0 \ H 7 0 0 0 0 z 0 H 8 Z z 0 ZfflZ z Z®ZfflZ@Z H 9 0 z Z 2 Z 2 z 4 H10 z 0 0 Z 2 H l l 0 z 0 12 z Z 2 z @ z ® z H13 0 Z 2 H1H 0 0 H15 z 0 H16 z r 2.17 A s s e r t i o n : H r ( G 2 n ) = <Z f o r r even-and r 4 n, r 5 2n Z © Z f o r r = n even 0 otherwise. <2 2> • • • > G 2 5 c a n ^ e g e n e r a l i z e d e a s i l y . C o r o l l a r y : H r(G2) = Z f o r r even, 0 f o r r odd. The method used f o r G, 35 Table I I I . Low dimensional homology groups f o r G : K (Note: H r ( G k ) = H r ( - 2]<- 5 r +i) since the embedding G k + 1 -* G k covers a l l c e l l s of G k of dimension r + 1 or l e s s . A l s o , H r ( G r + i ) = H r ( G r + 2 ) = . . . = H r(G) f o r the same reason.) H o H l H 2 H 3 \ H 5 H 6 G l z 0 0 0 0 0 0 z 0 z 0 z 0 0 5 3 z 0 Z 2 0 z Z 2 0 % z 0 Z 2 0 z ® z z 2 z 2 ® z 2 S 5 z 0 Z2 0 z ® z 2 • Z 2®Z 2 5 6 z 0 Z 2 0 z ® z 2 z © z 2 © z 2 G ? z 0 Z 2 • • z 2 © z 2 © z 2 G z 0 z 2 0 z ® z 2 z 2 z 2 © z 2 © z 2 In the unoriented Grassmannians and Schubert v a r i e t i e s , the computations are much simpler as there are only h a l f as many c e l l s t o worry about and the boundary map i s much simpler. 36 Table IV. Homology f o r the unoriented Grassmannians G, f o r s m a l l K j i i k and n: G2,2 G2,3 G2,4 G2,5 G3,3 G3,4 Z Z Z Z Z Z H l Z 2 Z2 Z2 Z 2 Z 2 , Z 2 H 2 z 2 z 2 z 2 z 2 Z 2 Z 2 H 3 0 Z2 Z 2 Z 2 z 2 © z 2 \ z z z ® z 2 z © z 2 z z © z 2 H 5 Z 2 Z 2 z 2 © z 2 z © z 2 © z 2 H 6 0 Z 2 Z 2 Z 2 Z2*522 H y ' 0 Z 2 Z 2 z „ © z „ 2 2 H8 Z Z 0 Z©Z„ 2 H 9 Z2 z Z 2 ^ 2 H10 0 0 H l l Z2 H 1 2 0 Note th a t there i s Poincare D u a l i t y i n G 0 0 , G 0 ., and G 2,2 ' 2,4 °3 s3 ' This r e f l e c t s the f a c t t h a t G j ^ i s o r i e n t a b l e whenever k + n i s even (see [ 2 ] ) . Remark: The homology groups f o r G k > n have been determined i n [ 7 ] , but as t h i s a r t i c l e was not a v a i l a b l e i n Russian or E n g l i s h i t was not 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 the 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 the Z 2 homology and cohomology products i n the unoriented Grassmannians are s t u d i e d , but the 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 formulas d e s c r i b i n g the cup product are equivalent v i a Poincare d u a l i t y (described i n terms of Schubert symbols i n 3.7) t o those d e s c r i b i n g products i n Z cohomology of G^ n ( C ) (see [ 3 ] ) . In the form g i v e n , they can a l s o be used t o determine cup products i n the unoriented Schubert v a r i e t i e s , and some examples are given. The' General I n t e r s e c t i o n Theory To Be Used For M a manifold, there i s a product theory f o r i n t e r s e c t i o n s of c y c l e s i n H t(M; Z^) c a l l e d the Lefschetz i n t e r s e c t i o n n L : H n - a ( M ' Z2> * H n - b ( M ' Z2> - H n - a - b ( M ' Z2> which i s r e l a t e d t o the cup product i n cohomology i n the f o l l o w i n g way. 3.1 A s s e r t i o n : For b = n - a above, the product fi^ induces a map D : H n_ a(M; Z 2) * H a(M; Z 2) Z H Q(M; Z 2) which can be considered as a map D : H n_ a(M; Z 2) -* Hom(Ha(M; Z 2) Z 2) w H 3(M; Z £) by a -y the map f ( B ) = a fl B € Z I B* L afl LB=l I f a n L p = Y i n H A(M; Z ) then D(a) u D(B) = D(Y) i n H*(M; Z 2 ) . This i s due t o the Lefschetz i n t e r s e c t i o n product being Poincare dual t o the cup product i n the sense that the f o l l o w i n g diagram commutes: 38 ( w i t h Z 2 c o e f f i c i e n t s ) (M) x H"( n [Ml D y a+b. . H (M) n [M] H n_ a(M) x H n_ b(M) -*H n-a-b [M] \ (M) where <-> [M] i s the cap product w i t h the fundamental c l a s s of M ( l y i n g i n H Q ( M ; Z 2 ) ) i . e . the Poincare dual map. 3.2 Ass e r t i o n : ' Let M be an n dimensional manifold w i t h a c e l l complex s t r u c t u r e , and e a , e^ c e l l s i n the complex r e p r e s e n t i n g Z 2 c y c l e s a and B. Suppose there i s continuous map h : M -> M homotopic to the i d e n t i t y such that f o r any c e l l s e a , c e a and e^, c ep , e Q , i s transverse t o Me^,). Then e a fl h(ep) i s a Z 2 c y c l e i n M homologous t o a fl^ B. I f e"a fl h ( i p ) = 0 then a n L B'= 0. This i s from general i n t e r s e c t i o n theory (e.g., see [10]). Simple I n t e r s e c t i o n s i n G, and the Poincare 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 of 3.2 i n t o Schubert c e l l terminology i s given below (3.4) and, using i t , the Poincare d u a l i t y map i s described i n terms of Schubert symbols (3.7) and some examples of e x p l i c i t i n t e r s e c t i o n s are given. 3.3 Remark and Nota t i o n : From the chain complex C^G^ n ) as s o c i a t e d with the Schubert c e l l decomposition (1.22) we obta i n a mod 2 chain complex C r ( G k n ; Z 2) and a mod 2 cochain complex C r ( G k } n ; Z 2) = Hom(C r(G k 5 n; Z 2 ) , Z 2 ) . For e 0 € C r ( G k n ; Z 2 ) , CT a Schubert symbol, w r i t e the cochain element dual to e C T as 0 " & j i . e . , a* € Hom(C ( G k n ; Z 2 ) , Z 2) i s the l i n e a r map sending eQ t o 1 and e^ t o 0 f o r r\ ± o. Since the cochain map 6 i s zero mod 2, H r ( G k > n ; Z 2) = C r ( G k j n ; Z 2) and {a* : \a\ = r } i s a b a s i s f o r H r(G k n ; Z 2) dual t o the b a s i s {e C T : |CT| = r } f o r H r ( G k j n ; Z 2 ) . From here on, Z 2 homology and cohomology w i l l be assumed unless otherwise s t a t e d . 3.4 Theorem: Let e^ and e^ be c e l l s i n G k n f o r a and r\ Schubert symbols. Suppose there i s an orthogonal l i n e a r t ransformation £ „k+n „k+n „ , * : R -»• R inducing $ : G k n G k n s u c n t h a t i ) For any e C T l c e a and , c e^ , e a t i s transverse t o # ( e v ) . i i ) e C T n * ( e ^ ) = ^ e ^ D U $3^7 (2 ) U . . . U * m e C T ( m ) f o r some orthogonal transformations 3^, . . . , § m , where cr(l) . . . o(m) are d i s t i n c t Schubert symbols of rank \o\ + |t]| - kn. Then i n H A(G k n ) , eQ 0^ e^ (the Lefschetz i n t e r s e c t i o n product) i s e C T ( 1 ) + e C T ( 2 ) + . . . + e a ( m ) . pf : This f o l l o w s from 3.2 as * : G k n G k n ^ s homotopic t o the i d e n t i t y , and ^ i e C T ( i ) i s homologous to e c r ( i ) ( s i n c e ^ i s a l s o homotopic t o the i d e n t i t y map). 3.5 Notation: Define S P : R q R Q, p 5 q by " ^ ( e ^ =J e p - i + l f o r i 2 p e^ f o r i > p. 40 I f q - k + n, then : G^ n -+ Gy. _ i s the induced homeomorphism. 3.6 Lemma: Let e C T and e„ be Schubert c e l l s i n G,, „ and <£> = § e^ D  bcnubert c e l l s i n ^ n k+n as defined above. Then we have i ) e„ i s transverse to i>e CT T] i i ) e C T fl e^ = 0 unless CT^ + T l ^ . i + i - n V i . i i i ) i f r)^ = n - CTk_^+1 V i then e C T fl $ = the p o i n t {P C T} i n G, . ( R e c a l l 1.6.) pf: i ) Suppose cr and r\ are Schubert symbols such t h a t CT^ + "H^.i+l - n - 1 f o r some i . I f P € e C T D <£> e^ , then dimension of P fl ](R ^ ) = i ( j as i n 1.4) and - "Hi, 1 - + 1 + k - i + 1 dimension of P D l ( R K ~ x + 1 ) = k - i + 1 cr.+i „ T j , . + 1 + k - i + l =» J(R 1 ) n K R k 1 + 1 ) > i - CTi + i + T i k _ i + 1 + k - i + l > n + k or c^ + > n - 1 which i s a c o n t r a d i c t i o n . Thus e C T fl = 0 . i i ) Suppose CT and rj are Schubert symbols such that CT^ + T ) - k _ i + 1 > n f o r a l l i . Consider the graph coordinates centred at P C T (see 1.17, 1.18 and 1.19). Since e^ c u C T , the i n t e r s e c t i o n e C T H ^ e ^ l i e s e n t i r e l y i n UCT , the domain of the graph coordinates. R e c a l l that L C T c R k x n £ s ^he l i n e a r subspace corresponding t o e C T . 'Claim: UCT D# e^ = (L_ f i f e - ) x La c R k x n i n the above graph coordinates. p f : L C T = {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^ c R . The plane cpCT(A) = P i s the row space of the matrix i n 1.18 corresponding t o A. Since P i s i n # e^ , i t must s a t i s f y the Schubert c o n d i t i o n s dimension of p'n i ( R ^ ) = i and dimension of P f | l ( R T l i + : L ~ 1 ) = i - 1. Looking at the matrix i n 1.18, i t can be seen that these c o n d i t i o n s are independent o f a^j f o r j > CT^ since r ) ] < _ i + i + °i - n-Thus P € * e^ fl U, CT « P = <PCT(A) f o r some A = ( a ^ J s . t . f o r A' = ( a l ^ ) where a i j = { a i j f ° r j 5 CTi 0 f o r j > CT^ and A' € L fl * e . CT T) Thus $ e^ n UCT = CLCT fl $ e n) x L CT This c l a i m proves 3 . 6 ( i i ) s i n c e L C T fl $ e^ has dimensi on |TTJ | - (kn - |cr| ) and the i n t e r s e c t i o n i s transverse ( s i n c e L C T i s orthogonal to L Q and thus to LQ fl <£> e^). i i i ) Suppose CT and r\ are Schubert symbols such t h a t a i + T V i + l = n • V i-P € $ e^ fl e"CT o P 0 j ( R i + 1 ) i n dimension > i and - r ' v _ i 4 . i + k " i + 1 n+k-o---i+l . . . P D l ( R = R ) i n dimension k - l f o r a l l I « P 3<e"rr.> f o r a l l i u i 42 Remark: Although 3.6 above shows t h a t we can always make two c e l l s e CT and e^ i n G k n transverse ( i n the manner of 3 . 4 ( i ) ) using the orthogonal k+n — — transformation $ . However, although the i n t e r s e c t i o n e C T fl * e^ must be a c y c l e homologous i n H.,.(Gk n ) t o e a ep , i t i s not i n general a union of orthogonal transformations of Schubert c e l l s as r e q u i r e d i n 3 . 4 ( i i ) . 3.7 Theorem: In G k n , the i n t e r s e c t i o n product ("1^. s a t i s f i e s : i ) For Schubert symbols cr and r\, e C T f l L e_ = 0 unless CT^ + T j ^ . i + i - n V i . i i ) f l L : H n_ p x H r -* H Q ^ Z 2 i s the map (CT, TI) •+ i i i f • T ] I = n - o- k_ i + 1 V i 0 otherwise. L i i i ) The Poincare d u a l i t y ( i n v e r s e ) map D : H -»• H P i s n-r CT -> TI* where TI . = n - CT, . , „ V i . l k-i+1 pf: ( 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 si n c e i f | cr j + |T)| = n and T)^ = n - °"k_i+^ does not hold f o r a l l i then 3 such t h a t T I . + CT, . . 1 < n. ^0 k-iQ+1 ( i i i ) i s j u s t another way of saying ( i i ) . This r e s u l t i s equivalent to 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: The i n t e r s e c t i o n product i s unnatural i n the sense t h a t f o r f : M -> M' a continuous ( c e l l u l a r ) map, f A does not preserve the i n t e r s e c t i o n product. However, i n cohomology, the induced map f* does 4 3 preserve cup product, so th a t i n the Grassmannians we have the f o l l o w i n g : i ) For j : G k -*. G k , , n' > n, j as i n 1 . 4 , j * : H*(G k n,) H*(G k j n) i s the map a* -y cr* (as can be seen from 3 . 3 , 1 . 2 1 and 2 . 1 2 ) and j*(cr* u r\*) = a* u TI*. i i ) For 1 : G k > n -»• G R, , k' > k, 1 as i n 1 . 4 , 1 * : H*(G k, n ) - H*(G k n ) i s the map CT* -*• ( C T 1 ) * where CT' = ( o " n + k , _ k + 1 , Q - n + k , ^ k + 2 , . . . , c r n + k t ) (as can be seen from 3 . 1 , 1 . 2 1 and 2 . 1 2 ) and l* ( c r * U T|&) = (cr')* U ( T ) ' ) * (T) ' defined from T) as 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 inverses f o r the maps j * and 1 * defined as, ( j * ) " 1 : H*(G k f n) - H * ( G k f n . ) n <n' i s the map (CT*) -»• (J,(CT))* and ( l * ) " 1 : H f t ( G k j n ) - H * ( G k , f n ) k 5 k ' i s the map cr* ->- ( l f ( c r ) ) * . These maps are group homomorphisms ( a c t u a l l y monomorphisms) but do not i n general preserve cup product. We now go t o some s p e c i f i c examples of i n t e r s e c t i o n product 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 general more complicated t o v e r i f y than i n 3 . 6 , so f o r the remainder of the paper we w i l l assume the f o l l o w i n g . 44 3.9 Assumption: Let e Q and e^ be Schubert c e l l s i n G^ . n , and suppose there i s an orthogonal transformation <S : R k + n -»• R k + n such t h a t e a fl # (where § : n -»• G k n i s induced by <£>) i s a ( Z 2 ) c y c l e Y o f dimension |CT| + \r\\ - kn. Then i ) e C T i s transverse to <i> e i i ) there i s an orthogonal transformation I 1' : R k + n R k + n such t h a t $ = e^ and f o r any e^, c e a a n d e^ t c e^ , e^ ., i s transverse t o <£' e^ i i i ) The c y c l e Y i s e C T fl e^ . Note: ( i i ) =» ( i i i ) 3.10 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, the Schubert c o n d i t i o n s a s s o c i a t e d w i t h eQ i n G^  n are dimension of P fl j(R°i + 1) > i V i . I f CT = n, then the above Schubert c o n d i t i o n i s redundant m (sin c e every k-plane i n R k + n i n t e r s e c t s j ( R n + i ) i n dimension i ) and can be l e f t out. i ) In G 2 2 look at 12 1*1 L 12. I f we take the orthogonal transformation ^ : G 2 2 &2 2 ( s e e 3.5), then 12 and # 4(12) s a t i s f y 3.4(i) but not 3 . 4 ( i i ) , so we must use o a d i f f e r e n t t r a n s f o r m a t i o n . Take § : G 2 2 G2 2 ( r e c a H 3.5): 12" n #3(T2~) = {P € G 2 2 s . t . dim. P fl <e±, e"2> _. 1 and dim. P fl <e"2, e"3> > 1} which i s e a s i l y seen t o be X^ U X 2 where 45 X1 = {P € G 2 2 s . t . dim. P fl <e2> = 1} X 2 = {P € G 2 j 2 s-1"' p c ^ 1 * e"2, e"3>} . Since X2 = 11 and X^ = $(02~) where $ i s any orthogonal t r a n s f o r m a t i o n t a k i n g e^ t o e~2 , by 3.9 we have 12 12 = 02 + 11 i n G 2 >2 • i i ) In G 2 3 look at 13 D L 23. Here, we use : G 2 3 -> G 2 3 : 13" (1 #4(23~) = {P € G 2 > 3 s . t . dim. P fl <e"1, e"2> > 1 and dim. P fl <e 2, 6 3 , e.,> > 1} which i s X^ U X 2 where X 1 = {P € G 2 3 s . t . dim. P fl <e"2> = 1 } X 2 - {P € G 2 3 s . t . dim. P fl <e±, e~2> > 1 and P c <e^, e 2 , eg, e^>} Since X 2 = 12 and X^ = $(03) f o r <£ as i n ( i ) above, we have 13 f l L 23 = 12 + 03 i n G 2 3 . 4 i i i ) In G 3 3 look at 133 D L 233, using $ : % 3 G 3 3 : 133" n §4(233") = {P € G 3 3 s . t . dim. P fl <e±, e"2> > 1 and dim. P fl <e 2, eg, e^> > 1} which as i n ( i i ) above i s X^ U X 2 where i+6 X-L = {P € G 3 3 s . t . P (1 <e"2> = 1} X 2 = {P € G 3 3 s . t . dim. P 0 <e~1, e"^ > 1 and P c < e ; L, e 2 , e"3, e"4>} However, here X 2 = 123 and X 1 = #(033) f o r # as above. Thus i n G g 3 we have 133 f l L 233 = 123 + 033. i v ) In G 3 j 3 look at 123 (1 L 233. Let <$: R ->• R be an orthogonal transformation mapping <e^, e 2 , e 3> t o <e 2, e^, e^> and # the induced homeomorphism on G 3 3 . 123 fl #(233) = {P € G 3 3 s . t . dim. P 0 <e±, e 2> > 1, dim. P fl <e 2, e^, e^> > 1 and dim. P fl <e^, e 2 , e 3 , e^> > 2}. 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^ U X 2 U X3 where X1 = {P € G 3 3 s . t . dim. P (1 <e"2> = 1 and dim. P fl <e^f e 2 , e 3 , e ^ > 2} X 2 = {P € G 3 3 s . t . dim. P (1 <e 1, e 2> > 1 and dim. P D <e^, e 2 , e^> > 2} X 3 = {P € G 3 3 s . t . dim. P fl <e 1, e 2> > 1 and . P c <i"lt . . . , _" >}. Under s u i t a b l e orthogonal transformations #^  , # 2 , and # 3 , 47 X± = ^ ( 0 2 3 ) , X 2 = * 2(Tl3") and X 3 = # 3(T22). Thus 123 DT 233 = 023 + 113 + 122 i n H (G 0 v) In G 2 ^ look at 34 D L 24, using $ = * 5: 24 fl $ 34 = {P € G 2 > 4 s . t . dim. P fl j ( R 3 ) > 1 and dim. P fl <e 2, e 3 , e^, e^> > 1} which i s X U Y where X = {P € G 2 4 s . t . dim. P fl <e~2, i~3> > 1} and Y = {P € G 2 4 s . t . dim. P fl j ( R 3 ) > 1 and P c j ( R 5 ) } . X = 3>^(14) and Y = 23 f o r some orthogonal t r a n s f o r m a t i o n ^ . Thus 34 f l L 24 = 14 + 23. v i ) In G 3 3 look at 222 D L 033. By 3 . 7 ( i ) , 222 f l L 033 = 0 i n G 3 3 , but l e t us t r y to make a nonempty i n t e r s e c t i o n , using 3^: 222" fl $ 5(033) = {P € G 3 3 s . t . dim. P fl <e~5> = 1 and P c <e^, . . . , e s>}-This i s the c e l l §5(022"), but | 022 | = 4 whereas | 222 | + j 033 [ - 9 5 = 6 + 6 - 9 = 3 . Thus we cannot use 3.9, although 222 and $ (033) (the open c e l l s ) are t r a n s v e r s e , having empty i n t e r s e c t i o n . 3.11 Remark: Using 3 . 7 ( i i i ) and 3.1 we can r e w r i t e the above r e s u l t s as cup products i n cohomology: i ) In H*(G 2 2 ) , 01* u 01* = 11* + 02* 48 i i ) In H*(G 2 3), 02* u 01* = 12* + 03* i i i ) In H*CG3 3) the same i s t r u e i v ) In H*(G 3 3), 12* u 01* = 112* + 22* + 13* v) In H*(G 2 4 ) , 02* u 01* = 12* + 03*. Note: 01* u 01* = 11* + 02* must hold i n H*(G, „) f o r a l l K ,n k ^ 2 and n > 2, s i n c e there are no other Schubert symbols of dimension 2. Complicated I n t e r s e c t i o n s and the General 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 3.10 so t h a t 3.4 can be used—3.14 has such examples—and f o r the cases where i t i s not p o s s i b l e , a more complicated argument, such as the one developed below, i s needed. The examples i n 3,14 lead up to the main formula i n 3.16. 3.12 D e f i n i t i o n : For k 5 k' d e f i n e g : {subsets of G, _} -+ {subsets of G, , \ K ,n K ,n as X c G v -> {P € G v, such t h a t P contains a K ,n K ,n k-plane j ( P ' ) c j ( R k + n ) f o r some P' 6 X}. 3.13 Claim: I f X i s a c y c l e i n ^(Gfc. n ) homologous to cr(l) + . . . + o"(m) where cr(i) are Schubert symbols i = 1, . . . , m, then g(x) i s a c y c l e i n ^ ^ . ( k ' _ k ) n ^ G k ' n^ homologous t o CT'(I) + . . . + <j'(m) where CT' ( i ) = ( o ( i ) 1 , c r(i) 0, . . . , cr(i), , n, n, . . . , n) V v ' k 1 - k f o r i = 1, . . . , m. This w i l l not be proved, but i t s v a l i d i t y i s suggested by the case X = ^(oTTT) U * 2 (O T 2 ) ) U. . . U *m(c7TmT) f o r some orthogonal transformations 3^, . . . , $ m . Here, the c l a i m i s obviously t r u e . 49 3.14 Examples: i ) Look at 24 |"IL 24 i n G 2 4 , using $ = $ 5 : 24" fl $(24) = {P € G 2 4 s . t . dim. P fl <e±, e"2, e"3> > 1 and dim. P D <e 3, e^, e 5> > 1}. This i s X U Y where X = {P € G 2 }t| s . t . dim. P D <e"3> = 1} and Y = {P € 24 n $(24) s . t . P c <e l s . . . , e5>} = 23" 0 *(23> Although 23 and $(23) do not s a t i s f y the 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 2 ^ , when considered as c e l l s i n G they do (by 3.6). Thus 23*n $ (23") i n G 2 3 i s homologous t o 23 f l L 23 = 13 + 22 (by 3.11), so Y = j(2"3~n $(23~)) i n G 2 ^ i s homologous t o 13 + 22 a l s o , since j preserves homology c l a s s . X i s $'(04) f o r some orthogonal transformation $', so combining the homology c l a s s e s determined by X and Y we o b t a i n 24 fl $ 24 = X U Y i s a c y c l e i n G 2 ^ homologous t o 1 3 + 2 2 + 0 4 . By 3.9 then, 24 fl 24 = 13 + 22 + 04 i n H,(G 0 u ) . In cohomology, by 3 . 7 ( i i i ) , . t h i s reads as 0 2 * u 02» = 13* + 22* + 04*. i i ) In G^ ^ look at 2334 n.L 2444 usi n g $ = $ 7: 2334" n $ 244¥ = {P 6 G^ 4 s . t . dim. P fl j(R3) > 1 _ dim. P fl j ( R 5 ) > 2 dim. P fl j ( R 6 ) > 3 and dim. P fl <e 5, eg, e y > - 1}. This i s X U Y where 50 X = 2334 fl # 24~W fl j ( G 4 3 ) = 2333 ("I $ 2333, and Y = {P € 2334 fl $ 2444 s . t . dim. P fl <e"5, e~6> > 1}, since f o r P € 2334 fl § 2444, i f P fl <e~5, e~6> = 0, then dim. P D ] ( R 7 ) = dim. P fl j ( R 6 ) + dim. P (1 <e"5, e"6, e~7> = 4 so tha t P must be i n X. As i n ( i ) above, 2333 fl <i> 2333 i n 3 i s homologous t o 2333 f l L 2333 = 1333 + 2233 (by 3.11 and 3 . 7 ( i i i ) ) , so that i n G 4 4 a l s o 2333 fl # 2333 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. (This can be e a s i l y checked.) In G3 3 , 233 and § 133 s a t i s f y the 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, thus 233 fl *(133) i n G3 3 i s a c y c l e homologous to 123 + 033 (by 3 . 1 0 ( i i i ) ) . Thus, by 3.13, Y = g(233 fl * 133" c G3 3) i s a c y c l e i n G^ ^ homologous t o 1234 + 0334. Combining the homology c l a s s e s of X and Y we have 2334 f l L 2444 = 1333 + 2233 + 0334 + 1234 i n H t ( G 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 formulas (3.15 and 3.16) completely describe the cohomology r i n g s t r u c t u r e i n G as a r i n g generated by the Schubert cocycles (0, 0, . . . , 0 , a ) * over a l l i n t e g e r s a > 1. 3.15 Claim: Let CT = (CT^, . . . , c r k ) and r ) = ( 0 , 0, . . . , 0 , T ] k ) be Schubert c y c l e s i n G k n . i ) For j : G k j n - G k j n , , n' > n and 51 ( j *) as i n 3.8, we have ( J * ) - 1 ( C T * u T ] * ) = CT* u T } * . i i ) For 1 : G k n ^ G k i > n •, k' > k and ( l * ) - 1 as i n 3.8, we have ( l * ) _ 1 ( a * u T i * ) = ( C T ' ) * U On 1 )* where CT ' = (0, 0, . . . , 0, CT. , . . . , CTv ) and r\ - (0, 0, . . . , 0, rj, ), v v J 1 v , ^ k' - k n + k' - 1 This can be proved by going t o the i n t e r s e c t i o n product v i a D (3.7) and g e n e r a l i z i n g f o r ( i ) , the way i n which 3 . 1 0 ( i i ) and ( v i ) give the same answer i n cohomology (3.11) and f o r ( i i ) , the way i n which 3 . 1 0 ( i i ) and ( i i i ) give the same answer i n cohomology (3.11). -1 -1 Note: In general i t i s not tru e that ( j * ) and (1*) preserve cup p r o d u c t — s e e 3.19(iv) and ( v ) . 3.16 Claim: Let CT = (CT^ = 0, CT2 > 0, CT3, . . . , 0 " k ) and T) = (0, 0, . . . , 0, r ] k ) be Schubert symbols. In H*(G) we have cr* u r|* = Z ( C T ' ) * , summed over a l l cr' = (cr' , . . . , CT'), Schubert 1 K symbols of dimension |CT'| = |CT| + r ) k such t h a t CT^ 5 cr| 5 CK + 1 f o r • < CT' k - CTk • i = 1 , . . . , k - 1 and cr, 5 ,' I n d i c a t i o n of proof: For CT and T) as above, define 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 Z^cr' and Z2CT' where Z^ i s over CT' s . t . CT' = (a£, CT^ + 1, o'l + 1, . . . , o» + 1) 52 f o r a l l CT" i n the sum f o r ( 0 , CT2 - 1 , 0-3 - 1 , • • • , CTk - 1 ) ' ( 0 , 0 , , 0 , T ] K ) Z 2 i s over CT ' s . t . cr' = (cr^' + 1, crjj + 1 , , CT" + 1) k f o r a l l CT" i n the sum f o r ( 0 , C T 2 - 1 , CTg - 1 , . . , cr - 1 ) • ( 0 , 0 , , 0 , T K - 1). pf: For cr' i n the sum f o r cr • r\ t e i t h e r = cr 2 i n which case CT' i s i n Z 0 , or cr' < cr i n which case cr' i s i n . 2 1- 2 1 Conversely, f o r cr' i n Z^ or Z 2 , i t i s easy t o see th a t CT' s a t i s f i e s cr^ 5 CT.! 5 o~^+^ and |CT'| = |CT| + r| k . By 3.15, the cup product cr* u r\* can be taken i n H*(G k n ) without l o s i n g any terms. Go to the corresponding i n t e r s e c t i o n product i n H (G. ) v i a the Poincare d u a l i t y ( 3 . 7 ( i i i ) ) . & k ,n J From here we can g e n e r a l i z e the method and r e s u l t i n 3.14(11), and we get X i s homologous to the dual ( 3 . 7 ( i i i ) ) of Z* and Y i s homologous t o the dual ( 3 . 7 ( i i i ) ) of Z* . In t h i s way, 3.16 can be proved by i n d u c t i o n on |CT| and r\ , . Note: The above formula i n H*(G) holds i n H*(G, ( C ) ; Z) where K 5TI i t i s known as P i e r i ' s formula (see [3]). 3.17 Claim: In H*(G) we have, ( C T C T cr 1* 1 1 ' 2' • * • ' °k ; CT(CTk)* CT(CT, CT(CTk + 1 ) : - 1 - D * °(Vl)ft CT(CT1 - k + 1 ) * CT(CT - k + 2 ) * . cr(crk + k - 1 ) * * °^CTk-l + k - 2 ) " CT(CT1)* 53 the determinant, where the product i s cup product, and cr(a) = (0, 0, . . . , 0, a) f o r a > 0 and 0 f o r a < 0. I n d i c a t i o n of proof: This 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 the s i z e o f the m a t r i x ; e.g., f o r k = 2 we have (CT1 , CT2 )* = O(CT2)5' CT(CT2 + I) --CT(CT1 - 1 ) * a ( a 1 ) * CT(CT9)" U CTCCT-, ) * + o-(o\. - 1 ) * u CT(CT0 + 1 ) * (cr-p o"2) + (CT1 - 1, CT2 + 1) + . . . + (o, CT2 + CT1) . + (0, o 2 + CT1) + (CT1 - 1, CT2 + 1) + (CT1 - 2, CT2 + 2) + . = (o±, a 2 ) (The second l i n e i s from 3.16.) Note: In H * ( G k n ( C ) , Z) t h i s i s c a l l e d the determinantal formula (see [ 3 ] ) . 3.18 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 the p r o j e c t i o n s 1* : H*(G) •+ H*(G K) and j * : H*(G k) -*-H*(G k > n). (See 3.8.) i i ) They are a l s o v a l i d f o r H*(S2(cr)) f o r any Schubert symbol CT, i f we use i * : H*(G, ) -»- H*(f2(a)) where K, n i : S2(CT) -* G, i s the embedding. K ,n 3.19 Examples using 3.18 above: The shortened n o t a t i o n i s used a g a i n , and f o r convenience we w i l l drop the 5' c's, as everything i s i n cohomology. 54 i i ) H*('S(1, 3)) cohomology products: 1 11 2 1 11 + 2 11 12 2 12 + 3 12 13 3 13 13 i i i ) H * ( l , 1, 3)) cohomology products: 1 11 2 1 11 + 2 11 111 + 12 0 2 12 + 3 112 + 13 13 112 112 0 113 12 112 + 13 113 113 3 13 113 0 112 113 13 113 113 | The next two examples are of i n d i v i d u a l cup products i n d i f f e r e n t Grassmannians. i v ) 124 u 2: In H*(G) i t i s 1224 + 1134 + 1125 + 234 + 225 + 144 + 135 + 126. In H*(G 3 4 ) i t i s 234 + 144. In H*(G 3 5) i t i s 234 + 225 + 144 + 135. In H*CG4 u ) i t i s 1224 + 1134 + 234 + 144; i . e . , f o r j : G 3 ^ ->• G 3 5 and 1 : G 3 ^ ->• G^ 4 (j 5 ' 0 and ( l 5 ' 0 ~ do not preserve cup product. v) 12 u 113 = (1 * 2 + 3) u 113: In H*(G) i t i s 11123 + 11114 + 1223 + 1133 + 1115 + 233 + 224 + 134 + 125. For j : G 3 > 3 - G g ^ and 1 : G 3 j 3 - G ^ —1 ' 1 and (1*) do not preserve cup product. Remark: In G 3 3 , 12 u 113 can be determined to be 233 (as above) by going t o the Poincare duals and using the i n t e r s e c t i o n method as i n 3.10. 3.20 Conclusion: The product 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 k) i s generated by the S-W cl a s s e s OJ^, . . . , of the t a u t o l o g i c a l bundle, and H"(G k n ) i s generated by co^, . . . , co^ . and co^, . . . , under the c o n d i t i o n s (1 + co^  + . . . + co k)(l + co1+ . . . + ccijj) = 1, where the Sj are the S-W c l a s s e s of the normal bundle. I t i s known (see [2]) t h a t COJ i s the cohomology c l a s s c r ( j ) * (from 3.17). By the map : G k n ->• G n k which i n cohomology must map «j t o C0j , we can f i n d the Schubert cocycle corresponding t o co^ : (cr( j ) ) = ( 1 , 1, . . . , 1) which we can c a l l """(j). V v J j times 56 Thus = co. , and T(.J)* must generate H*(G k). (This could 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 G, ( j ) ' i s obtained r e c u r s i v e l y from C f ( j ) by T ( j ) * = CT(j)* + CT(j - 1)* u T ( l ) * + + o(j - 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_^ + . . . + CO 1)(COJ+ Sj_ : L + . . . + o o ^ ) = 1 i n c h a r a c t e r i s t i c c l a s s e s . The above shows that f o r G, G k and G^ n , the cohomology r i n g has a simple d e s c r i p t i o n . However, i n the cohomology of Schubert v a r i e t i e s , the product s t r u c t u r e i s more complicated, and the simp l e s t method of d e s c r i p t i o n seems to be t o give a t a b l e f o r cup product as i n 3 . 1 9 ( i i ) and ( i i i ) . 57 References [1] J . W. M i l n o r and J . D , S t a s h e f f C h a r a c t e r i s t i c C l a s s e s , Annals of Mathematics Studies P r i n c e t o n U n i v e r s i t y P r e s s . 12] S. L. Kleiman Geometry of Grassmannians and a p p l i c a t i o n s . . . , Publ. Math. I . H. E. S. No, 36, P a r i s (1969). [3] S. L. Kleiman and D. Laksov Schubert C a l c u l u s , American Math Monthly, 79, pages 1061-1082 (1972), [4] W. V. D. Hodge and D. Pedoe Methods of 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 Press, 1953. [5] J . T. Schwartz D i f f e r e n t i a l Geometry and Topology, Gordon and Breach [6] S. S. Chern and Y u h - l i n Jou On the o r i e n t a b i l i t y of d i f f e r e n t i a b l e m a n i f o l d s , S c i . Rep. Nat. Tsing Hua Univ. 5, pages 13-17 (1948) [7] S. I . Al'ber Homologies of homogeneous spaces, Dokl. Akad. Nauk. USSR (N.S.) 98, pages 325-328, 1954 (Russian) 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 of 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. Lefschetz Topology American Math. So c i e t y Colloquium P u b l i c a t i o n s , Volume XII New York (1930)