BOUNDS FOR THE CUP-LENGTH OF
POINCAR'E SPACES AND THEIR APPLICATIONS
J'ulius Korba~s
Abstract. Our main result offers a new (quite systematic) way of deriving *
*bounds
for the cup-length of Poincar'e spaces over fields; we outline a general r*
*esearch program
based on this result. For the oriented Grassmann manifolds, already a limi*
*ted realization
of the program leads, in many cases, to the exact values of the cup-length*
* and to
interesting information on the Lyusternik-Shnirel'man category.
1. INTRODUCTION AND STATEMENT OF RESULTS
The cup-length over a field R, cupR (X), of a path connected topological spa*
*ce X is
the supremum of all numbers c such that there exist positive dimensional cohomo*
*logy
classes a1, . .,.ac 2 H*(X; R) such that their cup product a1 [ . .[.ac is nonz*
*ero.
Instead of the standard notation a [ b, we shall mostly write a . b or just ab.*
* Although
the main topic of this paper is the cup-length, we shall also keep in mind its *
*relation
to another important invariant: we have cat(X) 1 + cupR(X), where cat(X) is t*
*he
L-S category (the Lyusternik-Shnirel'man category). We recall that cat(X) is de*
*fined
to be the least positive integer k such that X can be covered by k open subsets*
* each
of which is contractible in X. If no such integer exists, then one puts cat(X) *
*= 1.
Note that some authors (see e.g. [5]) prefer to modify the definition by subtra*
*cting 1
from our value of cat(X).
The L-S category is very hard to compute; a longstanding problem in topology*
* (cf.
Ganea's list [7]) is the task to find its value for familiar manifolds. In gen*
*eral, the
cup-length is difficult to calculate, and (a formula for) its value remains unk*
*nown for
many commonly used spaces, e.g. for the great majority of the Grassmann manifol*
*ds
O(n)=O(k) x O(n - k); explicit formulae for cupZ2(O(n)=O(k) x O(n - k)) are kno*
*wn
for k 5, with some exceptions (cf. R. Stong's [12]).
1.1 Statement of the main result. When trying to find the cup-length of a space,
one might start from calculations of the height for some nonzero elements in its
cohomology. Recall that if x 2 Hi(X; R) (i > 0) is a nonzero cohomology class, *
*then
the height of x over R, denoted by htR(x), is the supremum of all the numbers k*
* such
that xk 6= 0; of course, one has cupR (X) htR(x). Our main result is the foll*
*owing
general theorem which we shall prove in Section 2. Its most interesting part, *
*(b),
shows that if X is an R-Poincar'e space, then height-related information may le*
*ad
_____________
2000 Mathematics Subject Classification. Primary: 57N65; 55M30 Secondary: 53*
*C30.
Key words and phrases. Cup-length; Lyusternik-Shnirel'man (Ljusternik-Schnir*
*elman) category;
Poincar'e space; flag manifold.
The author was supported in part by two grants of VEGA (Slovakia)
Typeset by AM S-T*
*EX
1
2 J'ULIUS KORBA~S
also to upper bounds for cupR (X); part (a) (in particular (a2) and (a3)) is al*
*most
obvious, but we include it in order to have the theorem as a comfortable refere*
*nce
tool.
Theorem A. Let R be a field, and X 6= ; be a path-connected R-Poincar'e space of
formal dimension n. Let the first two nonzero reduced R-cohomology groups of X,*
* in
dimensions less than n, be H"r(X; R) and H"q(X; R), r q < n. Then the cup-len*
*gth
satisfies the following.
(a) One always has
cup R(X) n_r. (a1)
If there is a cohomology class x 2 Hr(X; R) \ {0} such that r . htR(x) =*
* n,
then n
cup R(X) = __r. (a2)
If one finds a1, . .,.as 2 Hr(X; R)\{0} such that at11. .a.tss6= 0 for s*
*ome non-
negative integers t1, . .,.ts such that t1 + . .+.ts > 0 and r(t1 + . .+*
*.ts) < n,
then one has
cupR (X) 1 + t1 + . .+.ts. (a3)
(b) Suppose that r < q and there exists a basis a(1)r, . .,.a(t)r for the R-*
*vector
space Hr(X; R) such that
a(1)k1+1r= 0, . .,.a(t)kt+1r= 0
for some positive integers k1, . .,.kt satisfying the condition r(k1 + .*
* .+.kt) <
n.
Then the upper bound given in (a) improves to
~ ~
cupR (X) k1 + . .+.kt+ n_-_r(k1_+_._.+.kt)_q< n_r. (b1)
We remark that if X has the homotopy type of an (r - 1)-connected (r 1)
finite CW-complex, then, as proved by Grossman in [8], cat(X) 1 + dim_(X)_r; *
*as a
consequence, one then also has cupR (X) dim_(X)_r. Note that Grossman's inequ*
*ality
and our (a1) in Theorem A coincide for X having the homotopy type of an (r - 1)-
connected (r 1) finite CW-complex if X is an R-Poincar'e space; but our (a1)
requires just R-homological (r - 1)-connectedness, while for Grossman's upper b*
*ound
the standard (r - 1)-connectedness is required.
Theorem A can serve as a basis for the following research program.
Research Program. Let X be an R-Poincar'e space such as we suppose in the
theorem, such that r < q (if r = q, then one readily sees that cupR (X) = 2), a*
*nd such
that Hr(X; R) is finitely generated as a vector space over R. Then
(1) use (a1) to calculate the initial upper bound for cupR (X);
(2) study vanishing of products of elements in H*(X; R) and find (possibly u*
*sing
(a2) or (a3)) a lower bound, as high as you can, for cupR (X);
(3) if the best upper and lower bounds you have obtained do not coincide, th*
*en
find bases (as many as you can) in Hr(X; R), study vanishing of powers of
their elements and, when possible, apply (b1) to obtain a better upper b*
*ound.
BOUNDS FOR THE CUP-LENGTH AND THEIR APPLICATIONS 3
Note that this Research Program can be applied, in particular, to any R-orie*
*ntable
closed n-dimensional manifold with R-cohomology different from the R-cohomology
of the n-sphere Sn .
1.2 Statement of some applications of Theorem A. To illustrate its usefulness,
we shall use Theorem A, with r = 2, R = Z2 or r = 4, R = Q, for deriving several
new results on the cup-length of the oriented Grassmann manifolds; at the same *
*time,
we shall pay attention to their L-S category. In many cases, in spite of knowin*
*g the
cup-length, one might be very far from knowing the exact value of the L-S categ*
*ory
(see e.g. [5]). But we shall show that for the oriented Grassmann manifolds G"n*
*,k=
SO(n)=SO(k) x SO(n - k), with n 2k 6, the cup-length yields, at least in
some cases, a good amount of information about the L-S category (we shall take
k 3, because for G"n,1 = Sn-1 and for G"n,2(complex quadrics) the cup-length
and L-S category are known). More precisely, already our limited (in other wor*
*ds:
just illustrative) realization of the Research Program brings new non-trivial e*
*stimates
for the cup-length; in particular, we find the exact values of cupZ2(G"6,3) and*
* also of
cup Q(G"n,k) for infinitely many (n, k) with k 4. At the same time, it turns*
* out
that for G"n,3the cup-length lower bound (that is, 1 plus the cup-length) and t*
*he
L-S category can be very close to each other. In two cases, for G"6,3and G"7,3*
*, our
cup-length lower bound and the Grossman upper bound differ by just 1, for "G8,3*
*the
difference is 2.
Now we pass to detailed statements of the applications which we have roughly
outlined. In order to simplify the notation, we shall write cup(X) instead of c*
*upZ2(X),
and ht(x) instead of htZ2(x).
Using Theorem A(a) (mainly (a3); cf. (2) in the Research Program), we prove *
*the
following new lower bounds for the cup-length of the oriented Grassmann manifol*
*ds
in Section 3.
Proposition B. For the oriented Grassmann manifolds G"n,k, n 2k 6, we have
(a) cup (G"6,3) 3;8
>< n_+_3_ ifn 7 is odd,n =2{9, 11},
(b) cup (G"n,3) > 2
: n_+_2_ ifn 8 is even,n =2{10, 12}.
2
(c) Each of cup(G"9,3), cup(G"10,3), cup(G"11,3), cup(G"12,3) is at least 5.
(d) If k 4, then
8 n - k + 6
>>>_________ ifn - k + 3 7 is odd,n - k + 3 =2{9, 11},
< 2
cup(G"n,k) > n_-_k_+_5_
>>: 2 ifn - k + 3 8 is even,n - k + 3 =2{10, 12},
5 ifn - k + 3 2 {9, 10, 11, 12}.
(e) If k 4, then
8 ~ ~~ ~ ~ ~ ~ ~
>><1 + k_ n_-_k_ if 4 k_ n_-_k_ < k(n - k),
2 2 2 2
cupQ(G"n,k) > ~ ~ ~ ~ ~ ~ ~ ~
>: k_ n_-_k_ if 4 k_ n_-_k_ = k(n - k).
2 2 2 2
4 J'ULIUS KORBA~S
The author acknowledges that Proposition B(e) was suggested by Parameswaran
Sankaran. Proposition B obviously implies the following (the upper estimates ar*
*e the
Grossman upper bounds).
Corollary C. For the oriented Grassmann manifolds G"n,k, n 2k 6, we have:
(a) 4 cat(G"6,3) 5;
(b) n_+_5_2 cat(G"n,3) 3n_-_7_2if n is odd, n =2{9, 11}, and n_+_4_2 ca*
*t(G"n,3)
3n_-_7_if n is even, n =2{6, 10, 12}. In addition to this, we have 6
2
cat(G"9,3) 10, 6 cat(G"10,3) 11, 6 cat(G"11,3) 13, 6 cat(G"1*
*2,3)
14.
(c) If k 4, then
8
>>>6 if (n, k) = (8, 4),
>>< ~k ~~n - k ~ ~k~ ~n - k ~
__ ______ if 4 __ ______ < k(n - k),
cat(G"n,k) > 2 + 2 2 2 2
>>> ~ ~~ ~ ~ ~ ~ ~
>:1 + k_ n_-_k_ if (n, k) 6= (8, 4) and 4 k_ n_-_k_ = k(n*
* - k).
2 2 2 2
In Section 4, we shall prove the following upper bounds for cup(G"n,k) (n 6=*
* 6) and
cup Q(G"n,k) (k 4); the proofs can be seen as a realization of the third step*
* of our
Research Program. The manifold G"6,3will be treated in a special way; note that
for this manifold the upper bound coincides with the lower bound, so that we ob*
*tain
cup (G"6,3) = 3. Also note that the upper bound for cupQ (G"n,k) (k 4) given *
*below
coincides with the lower bound given in Proposition B(e) in infinitely many cas*
*es.
Proposition D. For the oriented Grassmann manifolds G"n,k, n 2k 6, we have
(a) cup (G"6,3) 3. As a consequence of this and Proposition B(a) , we have
cup (G"6,3) = 3.
(b) We have
8 ~ s+2 ~
>>> 2____-_7_ ifn = 2s + 1, s 3,
>>> 3
>>>~ s+2 ~
>>> 2____-_3_ s
>>< 3 ifn = 2 + 2, s 3,
cup(G"n,3) > ~ 2s+2 + 5 . 2p -~8 s p
>>> ________________ ifn = 2 + 2 + 1, s > p 1,
>>>~ 3 ~
>>> 2s+2 + 5 . 2p + 3t - 7
>>> ____________________ ifn = 2s + 2p + t + 1, s > p 1,
>: 3
1 t 2p - 1;
BOUNDS FOR THE CUP-LENGTH AND THEIR APPLICATIONS 5
8 ~ s ~
>>> 5_._2__-_13_ ifn = 2s + 1, s 3,
>>> 3
><
2s+1 - 4 ifn = 2s + 2, s 3,
cup(G"n,4) > s+1 s
>>>2 - 3 ifn = 2 + 3, s 3,
>>>~2s+1 + 4n - 17~
: _______________ if2s + 4 n 2s+1;
3
8 ~ s 2 ~
>>< (k_+_1)_._2__+_k_-_k__-_1 ifn = 2s + 1, s 3, k 5
3
cup(G"n,k) > ~ s+1 2 ~
>: 2____+_kn_-_k__-_1_ if2s + 2 n 2s+1, k 5;
3
(c) cup Q(G"n,k) k(n_-_k)_4if k 4. As a consequence of this and Proposi*
*tion
B(e) , we have cupQ (G"n,k) = k(n_-_k)_4for n even and k ( 4) even, as *
*well
as for n = 4t + 9 (t 1) and k = 4.
Note that Proposition B(d) gives cup (G"8,4) 5, while Proposition B(e) yie*
*lds
cup Q(G"8,4) 4 and, from Proposition D(c), we can see that cupQ (G"8,4) = 4. *
* This
indicates that, for a given "Gn,k, the value of cup(G"n,k) may be higher than c*
*upQ(G"n,k).
But the lower bounds presently known for the Z2-cup-length of "Gn,kwith k 4 a*
*nd
(n, k) 6= (8, 4), given in Proposition B(d), do not exceed the corresponding lo*
*wer
bounds for the rational cup-length, given in Proposition B(e).
We hope that our Research Program based on Theorem A can lead to further int*
*er-
esting results on the cup-length and L-S category not only for the oriented Gra*
*ssmann
manifolds but also for other manifolds.
2. ON THE CUP-LENGTH OF R-POINCAR'E
SPACES / PROOF OF THE MAIN RESULT
In the spirit of W. Browder's [4], by an R-Poincar'e space of formal dimensi*
*on n we
understand a path connected topological space X for which there is an element ~*
* 2
Hn(X) ~=R such that the \-product homomorphism \~ : Hk(X; R) ! Hn-k (X; R),
x 7! x \ ~, is an isomorphism for each k. By saying that an R-Poincar'e space X*
* is R-
homologically t-connected (t 0) we understand that its reduced cohomology gro*
*ups
H"i(X; R) vanish for all i t. For example, any t-connected (in the standard s*
*ense)
closed n-dimensional manifold orientable over R is an R-homologically t-connect*
*ed
R-Poincar'e space of formal dimension n.
2.1 Proof of Theorem A. If X is an R-Poincar'e space of formal dimension n,
then the cup product pairing Hk(X; R) x Hn-k (X; R) ! R is nonsingular (see e.g.
[4] or [9]); as a consequence, for any nonzero x 2 Hk(X; R) there exists some y*
* 2
Hn-k (X; R) such that x [ y is nonzero in Hn (X; R). In particular, this immedi*
*ately
implies (a3).
For the rest of the proof, first note that the hypothesis of Theorem A impli*
*es that
the space X is R-homologically (r - 1)-connected and it has at least three nont*
*rivial
unreduced R-cohomology groups. If there are just three, then Proposition A(a) *
*is
6 J'ULIUS KORBA~S
verified in an obvious way. Indeed, in such a case we have q = r = n_2, and one*
* readily
sees that cupR (X) = 2.
So suppose now that X has at least four nontrivial unreduced R-cohomology
groups, so that we have q > r. Of course (see the beginning of this proof), any
cup product of the maximum length, that is, of the length cupR (X), must belong*
* to
Hn (X; R) ~=R. So the cup-length of X is realized by a cup product
x(1)p1r. .x.(s)psryvqzj1q+i1.z.j.mq+im2 Hn (X; R) \ {0}, (o)
where x(1)r, . .,.x(s)r 2 Hr(X; R), yq 2 Hq(X; R), zq+il2 Hq+il(X; R) are nonze*
*ro
cohomology classes and p1, . .,.ps, v, j1, . .,.jm , i1, . .,.im are non-negat*
*ive integers.
Denote p = p1 + . .+.ps. Then, of course, cupR (X) = p + v + j1+ . .+.jm.
From this it is clear that
pr + vq + j1(q + i1) + . .+.jm(q + im ) = n,
therefore
n r(p + v + j1+ . .+.jm).
In other words, we obtain
n r . cupR (X),
which proves (a1). If there is a cohomology class x 2 Hr(X; R) \ {0} such that
r . htR(x) = n, then obviously cupR (X) n_r; this together with (a1) proves (*
*a2). The
rest of Theorem A(a) is clear in view of what we have said in the beginning of *
*this
proof.
We pass to the proof of Theorem A(b). Now we suppose that r < q, and we have
a basis a(1)r, . .,.a(t)r for Hr(X; R) such that
a(1)k1+1r= 0, . .,.a(t)kt+1r= 0
for some positive integers k1, . .,.kt such that r(k1 + . .+.kt) < n. As above,*
* take an
element realizing the cup-length of X, hence some
c := x(1)ss1r.x.(.s)sssryvqzj1q+i1.z.j.mq+im2 Hn (X; R) \ {0}, (o)
where x(1)r, . .,.x(s)r 2 Hr(X; R) \ {0}, yq 2 Hq(X; R) \ {0}, zq+il2 Hq+il(X; *
*R) \
{0}, and ss1, . .,.sss, v, j1, . .,.jmP, i1, . .,.im are non-negative integers*
*. We denote
ss = ss1 + . .+.sss. But now x(i)r = tj=1ffi,ja(j)r for some uniquely determ*
*ined
coefficients ffi,1, . .,.ffi,t2 R. So c is a linear combination of cup products*
* of the form
a(1)p1r. .a.(t)ptryvqzj1q+i1.z.j.mq+im, (oo)
where p1, . .,.pt are non-negative integers such that p1 + . .+.pt = ss. Since *
*c 6= 0,
at least one of the products (oo) must be nonzero; of course, in such a nonzero*
* cup
product, the exponent of a(i)r must be less than or equal to ki for all i = 1, *
*. .,.t.
We conclude that the cup-length is realized by a nonzero element
a(1)p1r. .a.(t)ptryvqzj1q+i1.z.j.mq+im2 Hn (X; R) \ {0}, (o o *
*o)
BOUNDS FOR THE CUP-LENGTH AND THEIR APPLICATIONS 7
where p1, . .,.pt, v, j1, . .,.jm, i1, . .,.im are non-negative integers, p1 *
*k1, . .,.pt
kt, cupR (X) = p1 + . .+.pt + v + j1+ . .+.jm, and yq 2 Hq(X; R) \ {0}, zq+il 2
Hq+il(X; R) \ {0}. Using the fact that q - r > 0, we obtain
n = r(p1 + . .+.pt) + q(v + j1 + . .+.jm ) + i1j1+ . .+.imjm
= q(p1 + . .+.pt+ v + j1 + . .+.jm ) + (r - q)(p1 + . .+.pt) + i1j1+ . .+.im*
*jm
q(p1 + . .+.pt+ v + j1 + . .+.jm ) + (r - q)(p1 + . .+.pt)
= q cupR(X) - (q - r)(p1 + . .+.pt)
q cupR(X) - (q - r)(k1 + . .+.kt).
The proof of Theorem A is finished.
3. APPLICATIONS: LOWER BOUNDS
FOR G"n,k/ PROOF OF PROPOSITION B
3.1 Selected facts about the oriented Grassmann manifolds. The oriented
Grassmann manifold "Gn,kconsists of oriented k-dimensional vector subspaces in *
*Rn,
similarly the Grassmann manifold Gn,k consists of all k-dimensional vector subs*
*paces
in Rn. For obvious reasons, we shall suppose that 2k n and, for the reason ex*
*plained
in the Introduction, we restrict ourselves to k 3.
Let fln,k (briefly fl) denote the canonical k-plane bundle over Gn,k, "fln,k*
*(briefly "fl)
be the canonical k-plane bundle over "Gn,k, and , be the canonical line bundle *
*associ-
ated with the double covering p : "Gn,k! Gn,k. We denote wi = wi(fl) in Hi(Gn,k*
*; Z2)
and "wi= wi("fl) in Hi(G"n,k; Z2) the corresponding Stiefel-Whitney classes. Of*
* course,
w"1= 0, and w1 is easily seen to coincide with the first Stiefel-Whitney class *
*of ,.
By Borel [2], the Z2-cohomology ring H*(Gn,k; Z2) can be identified with a q*
*uotient
ring,
H*(Gn,k; Z2) ~=Z2[w1, . .,.wk]=In,k,
where the ideal In,k is generated by the homogeneous components of
________1________
1 + w1 + . .+.wk
in dimensions n - k + 1, . .,.n.
Calculations in the ring H*(G"n,k; Z2) (k 3) may be very difficult. Fortun*
*ately,
they can be made a little easier thanks to the Gysin exact sequence associated *
*with
the double covering p : "Gn,k! Gn,k. Indeed, for showing that "wi22.w.".ikk6= 0*
*, it is
enough to verify that wi22. .w.ikk2 H2i2+...+kik(Gn,k; Z2) is not a multiple of*
* the class
w1.
In view of Theorem A (and in view of the step (3) of our Research Program), *
*it
is good to keep in mind an explicit description of the first nontrivial reduced*
* Z2-
cohomology group for those oriented Grassmann manifolds we are interested in.
Lemma E. For the oriented Grassmann manifolds G"n,k (n 2k 6), we have
H2(G"n,k; Z2) = {0, "w2} ~=Z2.
Proof. Let f : Vn,k ! G"n,k (where Vn,k is the Stiefel manifold of orthonormal *
*k-
frames in Rn) be the natural fiber bundle with fiber SO(k). The Lemma is readi*
*ly
implied by the corresponding Serre exact sequence.
8 J'ULIUS KORBA~S
3.2 Preparations for the proof of Propositions B(a)-(c). To any integer n 6,
we find s as the unique integer such that 2s < n 2s+1. Since dim(Gn,3) < 4n *
* 2s+3,
we have s+3
(1 + w2 + w3)2 = 1. (*)
To show that
wi22wi332 H2i2+3i3(Gn,3; Z2)
cannot be expressed as a multiple of the class w1, it is enough to show that wi*
*22wi33is
not in the ideal Jn,3 of Z2[w2, w3] generated by the homogeneous components of
_____1______= (1 + w + w )2s+3-1
1 + w2 + w3 2 3
(we have used (*)) in dimensions n-2, n-1, n. Therefore the ideal Jn,3is genera*
*ted by
homogeneous polynomials gn-2 , gn-1 , gn, obtained as the homogeneous components
in dimensions n - 2, n - 1 and n, respectively, of
2s+3-1XiX` i'
wi-j2wj3. (**)
i=0 j=0 j
2s+3-1
Note that the binomial coefficient i is odd for every i = 0, 1, . .,.2s+3*
* - 1. For
~ = n - 2, n - 1, n one calculates from (**) that
X ` i '
g~ = w3i-~2w~-2i3. (***)
~_3 i ~_23i - ~
3.3 Proof of Proposition B(a). From (***) we obtain that the ideal J6,3 in
Z2[w2, w3] is generated by g4 = w22and g6 = w23+ w32. So w2w3 is not in J6,3.
Consequently, w"2"w36= 0, and (cf. (a3) in Theorem A) cup (G"6,3) 3, as we h*
*ave
claimed.
3.4 Proof of Proposition B(b). We say that a homogeneous polynomial ha 2
Z2[w2,aw3],_in dimension a, is w2-monomial ("monomial" is an adjective here) if*
* ha =
w22. As a realization of the step (2) of our Research Program, we should find a*
* as high
as we can (this value of a will be called an available target dimension) such t*
*hat no
element in the a-dimensionalahomogeneous component of Jn,3 is w2-monomial. This
_
then gives that w"226= 0, and Proposition B(b) is readily implied by (a2) or (a*
*3) of
Theorem A.
Odd values of n. Let us first suppose that n is odd, n 7, n =2{9, 11}. We *
*shall
show that n + 1 is an available target dimension; so our aim now is to verify t*
*hat
none of the elements w3gn-2 , w2gn-1 , w3gn-2 + w2gn-1 is w2-monomial.
If n = 6t + 1, then using (***) we calculate that
w2gn-1 = w2w2t3+ . .+.w3t+12,
w3gn-2 = 0 . w2w2t3+ . .+.(3t - 1)w3t-22w23,
and
w2gn-1 + w3gn-2 = w2w2t3+ . .+.w3t+12.
BOUNDS FOR THE CUP-LENGTH AND THEIR APPLICATIONS 9
Hence none of w3gn-2 , w2gn-1 , w3gn-2 + w2gn-1 is w2-monomial, indeed.
If n = 6t + 3, we shall suppose that t > 1; the case n = 9 has a separate tr*
*eatment.
Write n = 3 . 2k+1 (2l + 1) + 3. Using (***), one readily verifies that none o*
*f the
polynomials w2gn-1 , w3gn-2 is w2-monomial. Further, one calculates that
X3t ` i + 1 '
w2gn-1 + w3gn-2 = w3i-6t-12w6t+2-2i3+ 1 . w3t+22.
i=2t+1 3i - 6t - 1
Therefore,
(a) if k 1 and l is arbitrary, then
k+1(2l+1) 3.2k(2l+1)+2
w2gn-1 + w3gn-2 = . .+.w22w23 + . .+.w2 ,
(b) if k = 0 and l is at least 2 and even, then
w2gn-1 + w3gn-2 = . .+.w52w4l3+ . .+.w6l+52,
(c) if k = 0 and l is at least 1 and odd, then
w2gn-1 + w3gn-2 = . .+.w6l+22w23+ w6l+52.
We conclude that for n = 6t + 3 with t > 1, the polynomial w2gn-1 + w3gn-2 is n*
*ot
w2-monomial.
For n odd, we are left with the case n = 6t + 5; we suppose that t > 1 (n = *
*11 is
treated separately). We write now n = 3 . 2k+1 . (2l + 1) - 1.
If l 1, then we obtain from (***) that
k 3.2k+2.l n+1_
w2gn-1 = w3.22w3 + . .+.w2 2 ,
n+1_ 3.2k(2l+1)-3
w3gn-2 = w3 3 + . .+.(3 . 2k(2l + 1) - 2)w2 w23.
n*
*+1_
Hence for l 1 none of the polynomials w2gn-1 , w3gn-2 , w2gn-1 + w3gn-2 = w3 *
*3 +
n+1_
. .+.w2 2 is w2-monomial.
We are left with n = 2k+1 . 3 . (2l + 1) - 1 for l = 0; now the argument use*
*d for
l 1 does not work. Note that we take k > 1, because we have n > 11.
Now one calculates from (***) that
n-2_ n-5_
gn-2 = w3 3 + . .+.0 . w2 2 w3.
Further, if k is odd, then
n+7_ n-11_ n+1_
w2gn-1 = . .+.w2 3 w3 9 + . .+.w2 2 ,
and if k is even, then
n+4_ n-5_ n+1_
w2gn-1 = . .+.w2 3 w3 9 + . .+.w2 2 .
Wenconclude+that1none of the polynomials w2gn-1 , w3gn-2 , w2gn-1 + w3gn-2 =
___ n+1_
w3 3 + . .+.w2 2 is w2-monomial.
Even values of n. Let us suppose that n is even, n 8, n =2{10, 12}. By a s*
*imilar
analysis as above, we could show that n is an available target dimension. But *
*the
hard work can be avoided. Indeed, we have the standard "inclusion" i : G"n-1,3!
G"n,3such that i*("fln,3) = "fln-1,3. Since n - 1 is odd, we know, by what we *
*have
computednabove, that n is an available target dimension for G"n-1,3. This impl*
*ies
_ n_
that "w222 Hn (G"n-1,3; Z2) does not vanish, and therefore "w222 Hn (G"n,3; Z2)*
* cannot
be zero. Since n is now less than dim (G"n,3), (a3) of Theorem A applies. Propo*
*sition
B(b) is proved.
10 J'ULIUS KORBA~S
3.5 Proof of Proposition B(c). To finish the proof of Proposition B, we are left
with the special cases, n = 9, 10, 11, 12. For n = 9, we obtain that g7 = w22w3*
*, g8 =
w2w23+ w42, g9 = w33. So "w426= 0, while "w52= 0 (indeed: we have w52= w3g7 + w*
*2g8).
In the three remaining cases, it is then readily seen that w"42is not zero (app*
*ly the
"inclusions" i : "G9,3! "G9+s,3such that i*("fl9+s,3) = "fl9,3, s = 1, 2, 3). P*
*roposition
B(c) is proved.
3.6 Proof of Proposition B(d). Let j : "Ga,b! "Ga+1,b+1be the "inclusion" such
that j*("fla+1,b+1) = "fla,b "1, where "1 is the trivial line bundle. Then we*
* have
j*(w2("fla+1,b+1)) = w2("fla,b). By an obvious iteration, we obtain the "inclu*
*sion" i :
G"n-k+3,3 ! G"n,ksuch that i*(w2("fln,k)) = w2("fln-k+3,3). In the proofs of P*
*roposi-
tions B(b), B(c), we have shown that certain powers of w2("fln-k+3,3) do not va*
*nish;
of course, then the same powers of w2("fln,k) do not vanish, and the rest is cl*
*ear: one
applies (a3) from Theorem A. Proposition B(d) is proved.
3.7 Proof of Proposition B(e). Now k 4 and, using the known description
of the cohomology algebra H*(G"n,k; Q) (cf. for instance [11; Proposition 5]),*
* one
readily verifies that we have r = 4 in Theorem A. Let pi("fln,k) 2 H4i(G"n,k; Q*
*) be the
ith rational Pontrjagin class. The height of p1("fln,k) is k_2 n-k_2(see e.g. *
*the proof
of [11; Theorem 1]). The lower bounds stated in Proposition B(e) are then impli*
*ed
by (a2) or (a3) of Theorem A.
The proof of Proposition B is finished.
4. APPLICATIONS: UPPER BOUNDS
FOR G"n,k/ PROOF OF PROPOSITION D
4.1 Proof of Proposition D(a). From the proof of Proposition B(a), we know that
w"2"w36= 0. Hence there exists a nonzero cohomology class a 2 H4(G"6,3; Z2) suc*
*h that
w"2. "w3. a 6= 0. It is clear that cup(G"6,3) 3, and that cup(G"6,3) could be*
* more than
3 only if a could be decomposed as a product of two cohomology classes (in view*
* of
Lemma E in 3.1, the only decomposition, not excluded a priori, would be a = "w2*
*2) or
if w"42would be nonzero. But the element a is indecomposable, because, in addit*
*ion
to the fact that H1(G"6,3; Z2) = 0, we have (Lemma E) H2(G"6,3; Z2) = {0, "w2},*
* and
w"22= 0 (cf. 3.3). So we conclude that cup(G"6,3) 3. Proposition D(a) is prov*
*ed.
4.2 Proof of Proposition D(b). We apply Theorem A(b), with r = 2, R = Z2. In
view of Lemma E, we have "w2as the only choice of basis in H2(G"n,k; Z2) (n 2*
*k 6).
Dutta and Khare [6] calculated the height of w2 in H*(Gn,k; Z2) as follows (we *
*quote
just the results we need, hence for n 2k 6).
Lemma F. (S. Dutta, S. Khare [6]) For the Grassmann manifolds Gn,k(n 2k 6)
BOUNDS FOR THE CUP-LENGTH AND THEIR APPLICATIONS 11
one has
8 s s
>>>2 - 1 ifn = 2 + 1,
>><2s ifn = 2s + 2,
ht(w2(fln,3)) = > 2s + 2p+1- 2 ifn = 2s + 2p + 1, s > p 1,
>>>s p+1 s p
>:2 + 2 - 1 ifn = 2 + 2 + t + 1, s > p 1,
1 t 2p - 1;
8 s s
>>>2 - 1 ifn = 2 + 1,
< 2s+1 - 4 ifn = 2s + 2,
ht(w2(fln,4)) = > s+1 s
>>:2 - 4 ifn = 2 + 3,
2s+1 - 1 if2s + 4 n 2s+1;
ae2s - 1 ifn = 2s + 1, k 5
ht(w2(fln,k)) =
2s+1 - 1 if2s + 2 n 2s+1, k 5.
It is clear that ht(w2(fln,k)), briefly ht(w2), cannot exceed half of the di*
*mension of
the corresponding Grassmann manifold. But, as we see from Lemma F, it is mostly
smaller. In an obvious way, the above quoted results on ht(w2) enable us to fi*
*nd,
for each pair (n, k) under consideration, some c, mostly smaller than half of t*
*he
dimension, such that "wc+12= 0.sFor instance, for G2s+1,3we know by Lemma Fsthat
ht(w2) = 2s - 1. Therefore w22 = 0, and of course for G"2s+1,3we have w"22 = 0.
Realizing the step (3) of our Research Program, using Theorem A(b), we obtain
the upper bound stated in Proposition D(b) for this case. The remaining cases *
*are
similar: when ht(w2) is less than half of the dimension, then we obtain the up*
*per
bound stated in Proposition D(b) by applying Theorem A(b), and in the cases whe*
*re
ht(w2) is precisely half of the dimension, we have just half of the dimension a*
*s an
upper bound for cup(G"n,k) from Theorem A(a).
4.3 Proof of Proposition D(c). The upper bound given in Proposition D(c) is
obtained from (a1) of Theorem A (with r = 4).
Proposition D is proved.
5. REMARKS
5.1 Remark. In a special case, for the real flag manifolds F (n1 + . .+.ns) = O*
*(n1 +
. .+.ns)=O(n1) x . .x.O(ns), we derived an upper bound of the same type as (b1)
of Theorem A, with r = 1 and R = Z2, in [10; Proposition 3.2.2]. We based it
there on specific properties of the Z2-cohomology of the flag manifolds, but we*
* did
not recognize the full potential, now expressed in Theorem A(b). In [10], we a*
*lso
illustrated the strength of that special case of (b1) by calculating the exact *
*value of
the Z2-cup-length of F (1, 2, n3) for all n3 3.
5.2 Remark. As pointed out by Akira Kono after having seen an early version of
the author's calculations for "Gn,3, one can deal with G2=SO(4) in a similar wa*
*y (note
that G2=SO(4) also is a simply-connected irreducible compact Riemannian symmetr*
*ic
space). Using Borel and Hirzebruch's [3; 17.3], one calculates that cup(G2=SO(4*
*)) =
4. As a consequence, cat(G2=SO(4)) 5. On the other hand, the Grossman upper
bound yields cat(G2=SO(4)) 5, and so cat(G2=SO(4)) = 5.
12 J'ULIUS KORBA~S
5.3 Remark. In view of Theorem A(b), it would be interesting to know the exact
values of ht(w"2).
5.4 Remark. For the oriented Grassmann manifolds "Gn,kwe know that 2 ht(w"2) <
k(n - k) whenever n is odd, independently of Dutta-Khare's Lemma F. Indeed, if n
*
* k(n-k)_
is odd, then (see e.g. [1; Theorem 1.1]) w2(G"n,k) = "w2. Hence the value of "w*
*2 2
on the fundamental class of the manifold "Gn,kis one of its Stiefel-Whitney num*
*bers.
But, as is well known, all the Stiefel-Whitney numbers of "Gn,kvanish.
Acknowledgement. The author thanks A. Kono, J. L"orinc, T. Macko, M. Mimura, P.
Sankaran, and R. Stong for their comments which were helpful at various stages *
*of
working on this paper.
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*elizability of Grassmann
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[7] Ganea, T., Some problems on numerical homotopy invariants, Lecture Notes in*
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[9] Hatcher, A., Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.
[10]Korba~s, J., L"orinc, J., The Z2-cohomology cup-length of real flag manifol*
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*113.
Department of Algebra, Geometry, and Mathematical Education, Faculty of Math-
ematics, Physics, and Informatics, Comenius University, Mlynsk'a dolina, SK-842*
* 48
Bratislava 4, Slovakia
E-mail address: korbas@fmph.uniba.sk
or
Mathematical Institute, Slovak Academy of Sciences, ~Stef'anikova 49, SK-814*
* 73
Bratislava 1, Slovakia
E-mail address: matekorb@savba.sk