Gröbner bases for (all) Grassmann manifolds
Zoran Z. Petrović, Branislav I. Prvulović, Marko Radovanović
aa r X i v : . [ m a t h . A T ] M a y GR ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS
ZORAN Z. PETROVI´C, BRANISLAV I. PRVULOVI´C, AND MARKO RADOVANOVI´C
Abstract.
Grassmann manifolds G k,n are among the central objects in ge-ometry and topology. The Borel picture of the mod 2 cohomology of G k,n isgiven as a polynomial algebra modulo a certain ideal I k,n . The purpose of thispaper is to understand this cohomology via Gr¨obner bases. Reduced Gr¨obnerbases for the ideals I k,n are determined. An application of these bases isgiven by proving an immersion theorem for Grassmann manifolds G ,n , whichestablishes new immersions for an infinite family of these manifolds. Introduction
Mod 2 cohomology of Grassmann manifolds G k,n = O ( n + k ) /O ( k ) × O ( n ) isthe polynomial algebra in Stiefel-Whitney classes w , . . . , w k of the canonical bun-dle over G k,n modulo the ideal I k,n generated by dual classes w n +1 , . . . , w n + k .Although the description of this ideal is simple enough, concrete calculations incohomology of Grassmann manifolds may be rather difficult to perform. The ques-tion of whether a certain cohomology class is zero is rather important in variousapplications — for example, in determining the span of Grassmannians, in dis-cussing immersions and embeddings in Euclidean spaces, in the determination ofcup-length (which is related to the Lusternik-Schnirelmann category), in some ge-ometrical problems which may be reduced to the question of the existence of anon-zero section of a bundle over a Grassmann manifold, etc. It is known thatGr¨obner bases are useful when one works with polynomial algebras modulo cer-tain ideal. The first use of Gr¨obner bases in this context appears in [9] where theGr¨obner bases for I ,n were established for n of the form n = 2 s − n = 2 s − I ,n and I ,n (for all n ) were estab-lished and used to obtain some new immersion results for Grassmann manifolds.At the time of writing of these papers, the authors were not aware of the paper[8], where additive bases for mod 2 cohomology of Grassmann manifolds were es-tablished. These additive bases together with information about Gr¨obner bases Mathematics Subject Classification. obtained directly in [12] and [13] allowed us to obtain reduced Gr¨obner bases forall I k,n .The plan of the presentation is as follows. In Section 2 some necessary factsabout cohomology algebra H ∗ ( G k,n ; Z ) are reviewed. Section 3 contains mainresults, namely, the determination of reduced Gr¨obner bases for all I k,n . Section 4is devoted to an application of the obtained results to the immersion problem for G ,n for n divisible by 8.2. The cohomology algebra H ∗ ( G k,n ; Z )In this section n and k are fixed integers such that n ≥ k ≥
2. Let G k,n = G k ( R n + k ) be the Grassmann manifold of k -dimensional subspaces in R n + k . Let γ k be the canonical vector bundle over G k,n and w , w , . . . , w k its Stiefel-Whitneyclasses. It is a direct consequence of Borel’s result ([2]) that the mod 2 cohomologyalgebra of G k,n is isomorphic to the polynomial algebra Z [ w , w , . . . , w k ] modulothe ideal I k,n generated by the dual classes w n +1 , w n +2 , . . . , w n + k . The followingequality holds for these dual classes:(1 + w + w + · · · + w k )(1 + w + w + · · · ) = 1 , and therefore, they satisfy the recurrence relation(2.1) w r + k = k X i =1 w i w r + k − i , r ≥ . Also, it is not hard to verify that the explicit formula for w r ( r ≥
1) is the following(see [13, p. 3]):(2.2) w r = X a +2 a + ··· + ka k = r [ a , a , . . . , a k ] w a w a · · · w a k k , where a , a , . . . , a k are understood to be nonnegative integers and [ a , a , . . . , a k ]denotes the multinomial coefficient[ a , a , . . . , a k ] = (cid:18) a + a + · · · + a k a (cid:19)(cid:18) a + · · · + a k a (cid:19) · · · (cid:18) a k − + a k a k − (cid:19) = k Y t =2 (cid:18)P kj = t − a j a t − (cid:19) . On the other hand, in [8] Jaworowski detected an additive basis for H ∗ ( G k,n ; Z ).Let us conclude this brief opening section by stating his result. Theorem 2.1 ([8]) . The set B = { w a w a · · · w a k k | a + a + · · · + a k ≤ n } is avector space basis for H ∗ ( G k,n ; Z ) ∼ = Z [ w , w , . . . , w k ] /I k,n . Gr¨obner bases
As usual, Z denotes the set of all integers. Recall that for α, β ∈ Z the binomialcoefficient (cid:0) αβ (cid:1) is defined by (cid:18) αβ (cid:19) := α ( α − ··· ( α − β +1) β ! , β > , β = 00 , β < , and therefore, the following lemma is straightforward. R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 3
Lemma 3.1. If (cid:0) αβ (cid:1) = 0 , then α ≥ β or α ≤ − . Recall also the well-known formula (which holds for all α, β ∈ Z )(3.1) (cid:18) αβ (cid:19) = (cid:18) α − β (cid:19) + (cid:18) α − β − (cid:19) . Let us now introduce some notations that we are going to use throughout thissection. For an integer m ≥ m -tuple N of integers we define the following m -tuples obtained from N (for i ≤ m and i < j ≤ m ): N i – by adding 1 to the i -th coordinate of N (if i <
1, then N i := N ); N i – by subtracting 1 from the i -th coordinate of N (if i <
1, then N i := N ); N i,j – by adding 1 to the i -th and j -th coordinate of N (if i <
1, then N i,j := N j ); N i,i – by adding 2 to the i -th coordinate of N (if i <
1, then N i,i := N ); N i,j – by subtracting 1 from the i -th and j -th coordinate of N (if i < N i,j := N j ); N i,i – by subtracting 2 from the i -th coordinate of N (if i <
1, then N i,i := N ).For an integer k ≥
2, a k -tuple A = ( a , a , . . . , a k ) and a ( k − M =( m , m , . . . , m k ) of integers, let: S A := k X j =1 a j , S ′ A := k X j =1 ja j , and S M := k X j =2 m j , S ′ M := k X j =2 ( j − m j ; P t ( A, M ) := (cid:18)P kj = t − a j − P kj = t m j a t − (cid:19) , for t = 2 , k ; P ( A, M ) := k Y t =2 P t ( A, M ).For example, P ( A, M ) = (cid:18) S A − S M a (cid:19) . Also, P ( A, ) = [ a , a , . . . , a k ], where = (0 , , . . . , | {z } k − ).Henceforth, the integers k and n with the property n ≥ k ≥ Z [ w , w , . . . , w k ]. Let us now define certain polynomialsin Z [ w , w , . . . , w k ] which will be important in our considerations. Definition 3.2.
For a ( k − M = ( m , . . . , m k ), let g M := X S ′ A = n +1+ S ′ M P ( A, M ) · W A , where the sum is taken over all k -tuples of nonnegative integers A = ( a , a , . . . , a k )such that S ′ A = n + 1 + S ′ M , and W A = w a w a · · · w a k k .Moreover, let G := { g M | S M ≤ n + 1 } . Note that, by (2.2), w n +1 = g ∈ G .Our aim is to prove that G is a Gr¨obner basis for I k,n = ( w n +1 , , . . . , w n + k )which determines the cohomology algebra H ∗ ( G k,n ; Z ). In order to do so, firstwe need to specify a term ordering in Z [ w , w , . . . , w k ]. We shall use the grlexordering (which will be denoted by (cid:22) ) on terms (monomials) in Z [ w , w , . . . , w k ] ZORAN Z. PETROVI´C, BRANISLAV I. PRVULOVI´C, AND MARKO RADOVANOVI´C with w > w > · · · > w k . It is defined as follows. The terms are compared bythe sum of the exponents and if these are equal for two terms, they are comparedlexicographically from the left. That is, for k -tuples A and B of nonnegative integerswe shall write W A ≺ W B if either S A < S B or else S A = S B and a s < b s where s = min { i | a i = b i } . Of course, W A (cid:22) W B means that either W A ≺ W B or W A = W B .In fact, we are going to prove that G is the reduced Gr¨obner basis for I k,n withrespect to the grlex ordering (cid:22) . We start with a lemma. Lemma 3.3.
If a k -tuple A = ( a , . . . , a k ) and a ( k − -tuple M = ( m , . . . , m k ) ofnonnegative integers are such that P ( A, M ) = 0 , then S A < S M or else P kj = t a j ≥ P kj = t m j for all t = 2 , k .Proof. Let us assume that S A ≥ S M . We will prove by induction on t that P kj = t a j ≥ P kj = t m j , for t = 2 , k . Since (cid:0) S A − S M a (cid:1) = P ( A, M ) = 0 and S A − S M ≥ S A − S M ≥ a , and therefore P kj =2 a j ≥ P kj =2 m j .Suppose now that P kj = t a j ≥ P kj = t m j for some t such that 2 ≤ t ≤ k −
1. Since P t +1 ( A, M ) = 0 and P kj = t a j ≥ P kj = t m j ≥ P kj = t +1 m j , again by Lemma 3.1 weconclude that P kj = t a j − P kj = t +1 m j ≥ a t . Hence, P kj = t +1 a j ≥ P kj = t +1 m j . (cid:3) For a nonzero polynomial f = P ri =1 t i ∈ Z [ w , w , . . . , w k ], where t i are pairwisedifferent terms, let T ( f ) := { t , t , . . . , t r } ( T (0) := ∅ ). The leading term of f = 0,denoted by LT( f ), is defined as max T ( f ) with respect to (cid:22) . Proposition 3.4.
Let M = ( m , . . . , m k ) be a ( k − -tuple of nonnegative integerssuch that S M ≤ n + 1 (i.e., such that g M ∈ G ). Then g M = 0 and LT( g M ) = W M ,where M = ( n + 1 − S M , m , . . . , m k ) . Moreover, if W A ∈ T ( g M ) \ { W M } for some k -tuple A of nonnegative integers, then S A < n + 1 .Proof. If we define m := n + 1 − S M , then obviously P t ( M , M ) = (cid:0) m t − m t − (cid:1) = 1, for t = 2 , k , and therefore P ( M , M ) = 1. Furthermore, S ′ M = k X j =1 jm j = n + 1 − S M + k X j =2 jm j = n + 1 + k X j =2 ( j − m j = n + 1 + S ′ M , and hence W M ∈ T ( g M ). So, g M = 0.Now take a k -tuple A = ( a , . . . , a k ) of nonnegative integers such that S ′ A = n + 1 + S ′ M and P ( A, M ) ≡ W A ∈ T ( g M ). Since S M = n + 1, inorder to finish the proof of the proposition, it suffices to show that if S A ≥ n + 1,then A = M .Since S M ≤ n + 1 ≤ S A , by Lemma 3.3 we have the following k − a k ≥ m k ,a k − + a k ≥ m k − + m k , ... a + · · · + a k ≥ m + · · · + m k . R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 5
Summing up these inequalities, we get k X j =2 ( j − a j ≥ k X j =2 ( j − m j . On the other hand, since S A ≥ n + 1 and S ′ A = n + 1 + S ′ M , k X j =2 ( j − a j = k X j =1 ( j − a j = S ′ A − S A ≤ S ′ A − ( n + 1) = S ′ M = k X j =2 ( j − m j , so all the inequalities in (3.2) are in fact equalities and S A = n + 1. Hence, a t = m t for t = 2 , k , and a = S A − P kj =2 a j = n + 1 − S M , i.e., A = M . (cid:3) Prior to the formulation of the following lemma, we would like to emphasize thatfor a ( k − M = ( m , . . . , m k ), by our definition, M i = ( m , . . . , m i +1 +1 , . . . , m k ), i = 1 , k −
1, and likewise for M i,j , M i,i , M i , etc. For example, the( k − M is defined as ( m , m +1 , . . . , m k ), and not as ( m +1 , m , . . . , m k ). Lemma 3.5.
Let A = ( a , a , . . . , a k ) be a k -tuple and M = ( m , . . . , m k ) a ( k − -tuple of integers. (a) For ≤ i ≤ j ≤ k − , P ( A, M i,j ) ≡ P ( A i , M j ) + P ( A, M i − ,j +1 ) + P ( A j +1 , M i − ) (mod 2) . (b) For ≤ i ≤ k − , P ( A, M i,k − ) ≡ P ( A i , M k − ) + P ( A k , M i − ) (mod 2) . Proof.
Let 1 ≤ i ≤ j ≤ k −
1. It is immediate from the definition that for all t = 2 , k , P t ( A, M i,j ) = (cid:18) a t − + a t + · · · + a k − m t − · · · − m k − δ t a t − (cid:19) , where δ t = , t ≤ i + 11 , i + 2 ≤ t ≤ j + 10 , t > j + 1 . Also, if t = i + 1, then P t ( A i , M j ) = (cid:18) a t − + a t + · · · + a k − m t − · · · − m k − δ t a t − (cid:19) , and so,(3.3) P t ( A, M i,j ) = P t ( A i , M j ) , for t = i + 1 . Likewise, using formula (3.1) we get(3.4) P i +1 ( A, M i,j ) + P i +1 ( A i , M j ) = P i +1 ( A, M j ) , since the left-hand side is equal to (cid:18) a i + · · · + a k − m i +1 − · · · − m k − a i (cid:19) + (cid:18) a i + · · · + a k − m i +1 − · · · − m k − a i − (cid:19) and the right-hand side to (cid:18) a i + · · · + a k − m i +1 − · · · − m k − a i (cid:19) . ZORAN Z. PETROVI´C, BRANISLAV I. PRVULOVI´C, AND MARKO RADOVANOVI´C (a) In this case, similarly as for (3.3) and (3.4), one obtains the following equal-ities: P t ( A i , M j ) = P t ( A, M i − ,j +1 ) , for t
6∈ { i + 1 , j + 2 } (3.5) P t ( A, M i − ,j +1 ) = P t ( A j +1 , M i − ) , for t = j + 2(3.6) P i +1 ( A, M j ) = P i +1 ( A, M i − ,j +1 )(3.7) P j +2 ( A i , M j ) = P j +2 ( A, M i − ,j +1 ) + P j +2 ( A j +1 , M i − ) . (3.8)So, using identities (3.3)–(3.8), we have P ( A, M i,j ) = k Y t =2 P t ( A, M i,j ) = P i +1 ( A, M i,j ) · k Y t =2 t = i +1 P t ( A i , M j ) ≡ (cid:0) P i +1 ( A i , M j ) + P i +1 ( A, M j ) (cid:1) · k Y t =2 t = i +1 P t ( A i , M j )= k Y t =2 P t ( A i , M j ) + P i +1 ( A, M i − ,j +1 ) · k Y t =2 t = i +1 P t ( A i , M j )= P ( A i , M j ) + P j +2 ( A i , M j ) · k Y t =2 t = j +2 P t ( A, M i − ,j +1 )= P ( A i , M j ) + (cid:0) P j +2 ( A, M i − ,j +1 ) + P j +2 ( A j +1 , M i − ) (cid:1) k Y t =2 t = j +2 P t ( A, M i − ,j +1 )= P ( A i , M j ) + P ( A, M i − ,j +1 ) + P ( A j +1 , M i − ) (mod 2) . (b) In a similar manner as before, for 1 ≤ i ≤ k − P t ( A i , M k − ) = P t ( A k , M i − ) , for t = i + 1 , (3.9) P i +1 ( A, M k − ) = P i +1 ( A k , M i − ) . (3.10)Now, using identities (3.3)–(3.4) and (3.9)–(3.10), we have P ( A, M i,k − ) = k Y t =2 P t ( A, M i,k − ) = P i +1 ( A, M i,k − ) · k Y t =2 t = i +1 P t ( A i , M k − ) ≡ (cid:0) P i +1 ( A i , M k − ) + P i +1 ( A, M k − ) (cid:1) · k Y t =2 t = i +1 P t ( A i , M k − )= P ( A i , M k − ) + P ( A k , M i − ) (mod 2) , and we are done. (cid:3) Note that we could unify parts (a) and (b) of the previous lemma by stating that P ( A, M i,j ) ≡ P ( A i , M j ) + P ( A j +1 , M i − ) + P ( A, M i − ,j +1 ) (mod 2), for 1 ≤ i ≤ j ≤ k −
1, with convention that P ( A, M i − ,j +1 ) = 0 if j = k − R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 7
Proposition 3.6.
Let M = ( m , . . . , m k ) be a ( k − -tuple of nonnegative integersand ≤ i ≤ j ≤ k − . Then in the polynomial algebra Z [ w , w , . . . , w k ] , we havethe identity g M i,j = w i g M j + w j +1 g M i − + g M i − ,j +1 , where the polynomial g M i − ,j +1 is understood to be zero if j = k − .Proof. By Lemma 3.5 we have g M i,j = X S ′ A = n +1+ S ′ Mi,j P ( A, M i,j ) · W A = X S ′ A = n +1+ S ′ Mi,j (cid:0) P ( A i , M j ) + P ( A j +1 , M i − ) + P ( A, M i − ,j +1 ) (cid:1) · W A = X S ′ A = n +1+ S ′ Mi,j P ( A i , M j ) · W A + X S ′ A = n +1+ S ′ Mi,j P ( A j +1 , M i − ) · W A + g M i − ,j +1 , since S ′ M i,j = S ′ M + i + j = S ′ M + i − j + 1 = S ′ M i − ,j +1 (for j ≤ k − S ′ A = n + 1 + S ′ M i,j is equivalent to S ′ A i = S ′ A − i = n + 1 + S ′ M i,j − i = n + 1 + S ′ M + j = n + 1 + S ′ M j , and likewise, it is equivalent to S ′ A j +1 = n + 1 + S ′ M i − .Now, consider the first sum in the upper expression. Since the sum is takenover the k -tuples A = ( a , a , . . . , a k ) of nonnegative integers (such that S ′ A = n + 1 + S ′ M i,j ), the coordinates of A i are also nonnegative with exception thatits i -th coordinate might be − a i = 0). But, in that case, P i +1 ( A i , M j ) = (cid:0) a i +1 + ··· + a k − m i +1 −···− m k − − (cid:1) = 0, and so P ( A i , M j ) = 0. Therefore, we may as-sume that a i ≥
1, and consequently, that A i runs through the set of k -tuples ofnonnegative integers (such that S ′ A i = n + 1 + S ′ M j ). Hence, X S ′ A = n +1+ S ′ Mi,j P ( A i , M j ) · W A = w i X S ′ Ai = n +1+ S ′ Mj P ( A i , M j ) · W A i = w i g M j . So, we are left to prove that the second sum in the upper expression for g M i,j isequal to w j +1 g M i − . Let A = ( a , a , . . . , a k ) be a k -tuple of nonnegative integerssuch that S ′ A = n + 1 + S ′ M i,j , i.e., S ′ A j +1 = n + 1 + S ′ M i − . It suffices to show that a j +1 = 0 implies P ( A j +1 , M i − ) = 0, since then the proof follows as for the firstsum.If j +1 < k , then a j +1 = 0 implies P j +2 ( A j +1 , M i − ) = 0, and so, P ( A j +1 , M i − ) =0. For j = k −
1, let us assume to the contrary that a k = 0 and P ( A k , M i − ) = 0.First we shall prove that(3.11) a t − + a t + · · · + a k − ≤ m t + · · · + m k + ε t , for all t = 2 , k ,where ε t = (cid:26) , ≤ t ≤ i , i + 1 ≤ t ≤ k . The proof is by reverse induction on t . For theinduction base we prove (3.11) for t = k . Since (cid:0) a k − − − m k a k − (cid:1) = P k ( A k , M i − ) = 0and a k − − − m k < a k − , by Lemma 3.1 we conclude that a k − − − m k ≤ −
1, so a k − ≤ m k = m k + ε k . For the inductive step, let 2 ≤ t ≤ k −
1, and suppose that
ZORAN Z. PETROVI´C, BRANISLAV I. PRVULOVI´C, AND MARKO RADOVANOVI´C a t + · · · + a k − ≤ m t +1 + · · · + m k + ε t +1 . Since obviously ε t +1 ≤ ε t , we actuallyhave that a t + · · · + a k − ≤ m t +1 + · · · + m k + ε t . Since P t ( A k , M i − ) = (cid:18) a t − + a t + · · · + a k − − − m t − m t +1 − · · · − m k − ε t a t − (cid:19) = 0 , and a t − + a t + · · · + a k − − − m t − m t +1 − · · · − m k − ε t ≤ a t − − − m t < a t − ,according to Lemma 3.1, we have that a t − + a t + · · · + a k − − − m t − m t +1 −· · · − m k − ε t ≤ −
1, i.e., a t − + a t + · · · + a k − ≤ m t + m t +1 + · · · + m k + ε t .Now, summing up inequalities (3.11), we get S ′ A ≤ S ′ M + k X t =2 ε t < S ′ M + k − < S ′ M + n + 1 ≤ S ′ M i − + n + 1 = S ′ A k , which is a contradiction since S ′ A > S ′ A − k = S ′ A k . (cid:3) In the following lemma we establish a connection between polynomials g M andpolynomials (dual classes) w r ∈ Z [ w , w , . . . , w k ] from the previous section. Lemma 3.7.
For m ≥ and ( k − -tuple M = ( m, , . . . , we have that g M = m X i =0 (cid:18) mi (cid:19) w m − i w n +1+ i . Proof.
The polynomials g M were introduced in Definition 3.2 and they depend onthe (previously fixed) integer n . In this proof (and only in this proof) we allow n to vary through the set { k, k + 1 , . . . } , while the integer k ≥ Z [ w , w , . . . , w k ]). Note that the polynomials w r , r ≥
1, are defined independently of n . We emphasize the dependence of g M on n by using an appropriate superscript, and we actually prove the following claim: g ( n ) M = m X i =0 (cid:18) mi (cid:19) w m − i w n +1+ i , for all m ≥ n ≥ k. The proof is by induction on m . We have already noticed that g ( n ) = w n +1 , andtherefore, the claim is true for m = 0 (and all n ≥ k ). So, let m ≥ m − n ≥ k . Let M = ( m, , . . . ,
0) and n ≥ k . Then M = ( m − , , . . . ,
0) and since, for all k -tuples A of integers, P ( A, M ) ≡ P ( A , M )+ P ( A, M ) (mod 2) by (3.1) and P t ( A, M ) = P t ( A , M ) = P t ( A, M ) R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 9 for t = 3 , k , we have that g ( n ) M = X S ′ A = n +1+ S ′ M P ( A, M ) · W A = X S ′ A = n +1+ S ′ M ( P ( A , M ) + P ( A, M )) · k Y t =3 P t ( A, M ) ! · W A = w X S ′ A = n +1+ S ′ M P ( A , M ) · W A + X S ′ A =( n +1)+1+ S ′ M P ( A, M ) · W A = w g ( n ) M + g ( n +1) M = w m − X i =0 (cid:18) m − i (cid:19) w m − − i w n +1+ i + m − X i =0 (cid:18) m − i (cid:19) w m − − i w n +2+ i = m X i =0 (cid:18) m − i (cid:19) w m − i w n +1+ i + m X i =0 (cid:18) m − i − (cid:19) w m − i w n +1+ i = m X i =0 (cid:18) mi (cid:19) w m − i w n +1+ i , and the proof is completed. (cid:3) Proposition 3.8. G ⊆ I k,n .Proof. Since the ideal I k,n is generated by the polynomials w n +1 , w n +2 , . . . , w n + k ,note that, by the recurrence relation (2.1), not only these k polynomials, but all w r for r ≥ n + 1 belong to I k,n . Likewise, we shall prove that g M ∈ I k,n for all( k − M of nonnegative integers, and not only for those with the property S M ≤ n + 1 (i.e., g M ∈ G ).We define the relation < lexr on the set of all ( k − a , a , . . . , a k − ) < lexr ( b , b , . . . , b k − ) ⇐⇒ a s < b s , where s = max { i | a i = b i } , which is exactly the strict part of the lexicographical right ordering. This is a wellordering and our proof is by induction on < lexr .For the ( k − M = ( m, , . . . , m ≥ g M ∈ I k,n . So, let us now take a ( k − M = ( m , m , . . . , m k ) such that thegreatest integer s with the property m s +1 > ≤ s ≤ k − M = ( m , . . . , m s +1 , , . . . , g M ′ ∈ I k,n for all M ′ such that M ′ < lexr M . We wish to prove that g M ∈ I k,n . By Proposition 3.6applied to the ( k − M s , i = 1 and j = s − g M = g M ,s − s + w g M s − s + w s g M s . Since M s < lexr M s − s < lexr M ,s − s < lexr M , we conclude that g M ∈ I k,n . (cid:3) In the following proposition we formulate a characterization of Gr¨obner baseswhich we shall use for the proof that the set G is a Gr¨obner basis for the ideal I k,n .The proof of the proposition can be found in [1, Proposition 5.38(vi)]. Proposition 3.9.
Let F be a field, F [ x , x , . . . , x k ] the polynomial algebra, and I an ideal in F [ x , x , . . . , x k ] . Suppose that a term ordering (cid:22) in F [ x , x , . . . , x k ] isfixed. Let G be a finite subset of I such that / ∈ G , and let B ⊆ F [ x , x , . . . , x k ] /I be the set of cosets of all terms which are not divisible by any of the leading terms LT( g ) , g ∈ G . Then G is a Gr¨obner basis for I with respect to (cid:22) if and only if B is a vector space basis for F [ x , x , . . . , x k ] /I . We are now finally in position to prove our main result.
Theorem 3.10.
The set G (see Definition 3.2) is the reduced Gr¨obner basis forthe ideal I k,n with respect to the grlex ordering (cid:22) .Proof. By Propositions 3.4 and 3.8, 0 / ∈ G ⊆ I k,n , and it is obvious from thedefinition that G is finite. Also, according to Proposition 3.4 again, the set { LT( g ) | g ∈ G } is exactly the set of all terms in Z [ w , w , . . . , w k ] with the sum of theexponents equal to n + 1, that is { LT( g ) | g ∈ G } = { W A | S A = n + 1 } . Therefore,the set of all terms which are not divisible by any of the terms in { LT( g ) | g ∈ G } isjust the set { W A | S A ≤ n } . By Proposition 3.9 and Theorem 2.1, G is a Gr¨obnerbasis for I k,n .Since { LT( g ) | g ∈ G } = { W A | S A = n + 1 } and all terms of g ∈ G except theleading one have the sum of the exponents at most n (Proposition 3.4), no term of g is divisible by any other leading term in G . This means that Gr¨obner basis G isthe reduced one. (cid:3) Propositions 3.4 and 3.6 enable us to explicitly determine the polynomials g M ∈ G for the ( k − M = ( m , m , . . . , m k ) such that m k is close to n . Namely,if g M ∈ G and W A ∈ T ( g M ) \ { W M } (where M = ( n + 1 − S M , m , . . . , m k )), then S A ≤ n by Proposition 3.4. Consequently, S ′ A = P kj =1 ja j ≤ k P kj =1 a j = kS A ≤ kn . On the other hand, S ′ A = n + 1 + S ′ M , and so, we conclude that g M = W M whenever S ′ M > ( k − n − N be the ( k − , . . . , , n ). Since S ′ N s > S ′ N = ( k − n (for s = 1 , k − g N = w w nk and g N s = w s +1 w nk , ≤ s ≤ k − . If we apply Proposition 3.6 to the ( k − N k − = (0 , . . . , , n − i = 1 and j = k −
1, we obtain the relation w k g N k − = g N + w g N . Both summands on theright-hand side contain w k as a factor, so w k cancels out and using (3.12) we get(3.13) g N k − = w w n − k + w w n − k . Likewise, by applying Proposition 3.6 to N k − , i = s + 1 and j = k −
1, one obtainsthat w k g N sk − = w s +1 g N + g N s +1 , and so(3.14) g N sk − = w w s +1 w n − k + w s +2 w n − k , ≤ s ≤ k − . Identities (3.13) and (3.14) determine g M ∈ G when m k = n − S M ≤ n . Forcomputing g M ∈ G when m k = n − S M = n + 1 for a concrete integer k , onecan use Proposition 3.6 and apply it first to N k − , i = 1 and all j = 1 , k −
2, thento N k − , i = 2 and all j = 2 , k − g M ∈ G for m k = n − k = 2 and k = 3, all the members g M ofGr¨obner basis G for m k ≥ n − k = 2) and [13] (for k = 3). R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 11
Since our application of Gr¨obner bases (given in the next section) treats the case k = 5, let us write down the relations (3.13) and (3.14) in this case:(3.15) g (0 , , ,n − = w w n − + w w n − ,g (1 , , ,n − = w w w n − + w w n − ,g (0 , , ,n − = w w w n − + w w n − ,g (0 , , ,n − = w w w n − + w n . Remark . Since the description of the mod 2 cohomology of the complex Grass-mann manifolds G k ( C n + k ) is essentially the same as the one in the real case (theonly difference being in the fact that dimensions of the Stiefel-Whitney classes aremultiplied by 2), it is immediate that the reduced Gr¨obner basis for the correspond-ing ideal in this case can be obtained from the Gr¨obner basis G (Definition 3.2) bysubstituting w i for w i ( i = 1 , k ) in all polynomials g M ∈ G .4. Application to immersions
In this section we consider the (real) Grassmannians G ,n , where n is divisibleby 8. As before, w i ∈ H i ( G ,n ; Z ), i = 1 ,
5, is the i -th Stiefel-Whitney class of thecanonical bundle γ over G ,n . Lemma 4.1.
Let n ≡ and let ν be the stable normal bundle over Grass-mann manifold G ,n . Then for the Stiefel-Whitney classes of this bundle, the fol-lowing equalities hold: w ( ν ) = w + w and w i ( ν ) = 0 when i ≥ n − .Proof. Let r ≥ r < n + 5 ≤ r +1 . Note that this implies n ≥ r since n ≡ w ( ν ) = w ( γ ⊗ γ ) · (1 + w + w + w + w + w ) r +1 − n − , and that the top nonzero class in this expression is in dimension 20+5(2 r +1 − n − n ≥ r , we have that 20 + 5(2 r +1 − n − ≤
20 + 5(2 r −
5) = 5 · r − ≤ n − w ( γ ⊗ γ ) = w ( γ ⊗ γ ) = 0, which is nothard to check by the method described in [10, Problem 7-C]. Using this fact and(4.1), one obtains that w ( ν ) = (cid:18) r +1 − n − (cid:19) w + (2 r +1 − n − w = w + w , since 2 r +1 − n − ≡ (cid:3) Theorem 4.2. If n ≡ , then G ,n immerses into R n − .Proof. Since dim G ,n = 5 n , in order to prove that there is an immersion of G ,n into R n − , by Hirsch’s theorem ([7, Theorem 6.4]) it suffices to show that theclassifying map f ν : G ,n → BO of the stable normal bundle ν over G ,n lifts up to BO (5 n −
3) . G ,n BO f ν / / BO (5 n − BO p (cid:15) (cid:15) G ,n BO (5 n − < < ③③③③③③ We shall use the method of modified Postnikov towers (MPT) introduced byGitler and Mahowald in [5] and extended to fibrations p : BO ( m ) → BO for m oddby Nussbaum in [11]. So, we factor out the map p : BO (5 n − → BO as indicatedin the following diagram and then we lift the map f ν one level at the time. Thediagram presents the 5 n -MPT for the fibration p ( K m stands for the Eilenberg-MacLane space K ( Z , m )). Also, the relations that determine the k -invariants ofthe tower are listed in the table. G ,n BO f ν / / BO K n − × K nk × k / / E BO (cid:15) (cid:15) E K n − × K nk × k / / E E (cid:15) (cid:15) E K nk / / BO (5 n − E (cid:15) (cid:15) G ,n E ♦♦♦♦♦♦♦ G ,n E ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ G ,n BO (5 n − D D ✡✡✡✡✡✡✡✡✡✡✡✡✡ k = w n − k = w n k : ( Sq + w ) k = 0 k : ( Sq + w + w ) Sq k + Sq k = 0 k : ( Sq + w ) k + Sq k = 0According to Lemma 4.1, w n − ( ν ) = w n ( ν ) = 0, so, we can lift f ν up to E .Also, since Sq ( w w n − ) = ( w w + w ) w n − + w ( n − w w n − = nw w w n − + w n = w n , and since w n = 0 in H n ( G ,n ; Z ) ∼ = Z (Theorem 2.1), by looking at the relationsin the table for k and k , we see that it is easy to overcome these two k -invariants.Therefore, the only obstruction left to deal with comes from the k -invariant k .Since H n − ( G ,n ; Z ) ∼ = Z , it suffices to show that the map (cid:0) Sq + w ( ν ) (cid:1) : H n − ( G ,n ; Z ) → H n − ( G ,n ; Z ) is nontrivial. We use Lemma 4.1, formulas ofWu and Cartan and the polynomials from (3.15) (which are trivial in H ∗ ( G ,n ; Z )since they are members of the Gr¨obner basis G for the ideal I ,n , and hence, they R ¨OBNER BASES FOR (ALL) GRASSMANN MANIFOLDS 13 belong to I ,n ) to calculate (cid:0) Sq + w ( ν ) (cid:1) ( w w n − ) = ( Sq + w + w )( w w n − )= w w n − + ( w w + w )( n − w w n − + w (cid:18) ( n − w w n − + (cid:18) n − (cid:19) w w n − (cid:19) + w w w n − + w w n − = w w w n − + w w w n − + w w n − = w g (0 , , ,n − + g (0 , , ,n − + w w n − = w w n − , and this class is nonzero by Theorem 2.1. (cid:3) By the famous result of Cohen ([3]), Grassmanian G ,n can be immersed into R n − α (5 n ) , where α (5 n ) denotes the number of ones in the binary expansion of 5 n .This means that Theorem 4.2 improves this result whenever α (5 n ) = 2 (and n ≡ n is a power of two, and it is known that then G ,n cannot be immersed into R n − ([6, p. 365]). So, if n is a power of two, thenfor imm( G ,n ) = min { d | G ,n immerses into R d } the following inequalities hold10 n − ≤ imm( G ,n ) ≤ n − . Actually, a sufficient and necessary condition for α (5 n ) = 2 and n ≡ n is of the form 2 r + P si =0 (2 r +2+4 i + 2 r +3+4 i ), r ≥ s ≥ − s = − n = 2 r ). References
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