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An explicit formula for the cup-length of the rotation group.

1 introduction and the main result

As is well known, the [Z.sub.2]-cup-length cup(X; [Z.sub.2]) of a compact path-connected topological space X is the maximum of all integers c such that there exist reduced cohomology classes [a.sub.1],..., [a.sub.c] [member of] [??]*(X; [Z.sub.2]) such that their cup product [a.sub.1] [union] *** [union] [a.sub.c] does not vanish. Instead of the usual notation a [union] b, we shall write ab, H*(X;[Z.sub.2]) will be abbreviated to H*(X), and cup(X;[Z.sub.2]) will be shortened to cup(X) in the sequel (we shall only consider cohomology with coefficients in [Z.sub.2]). The Elsholz inequality cat(X) [greater than or equal to] cup(X) relates cup(X) to another important homotopy invariant, the Lyusternik-Shnirel'man category cat(X); the latter is defined to be the least positive integer k such that X can be covered by k + 1 open subsets each of which is contractible in X.

For the rotation (or special orthogonal) group SO(n), the [Z.sub.2]-cohomology algebra is known due to A. Borel [1]. We recall its description by A. Hatcher [2]:

[mathematical expression not reproducible], (1)

where the degree of [[beta].sub.i] is equal to i and [p.sub.i] is the smallest power of 2 such that the degree of [mathematical expression not reproducible] is at least n. This cohomology algebra looks quite simple but, to the best of the author's knowledge, only recursive formulas for cup (SO (n)) were known up to now: the formula cup(SO(2n)) = 2cup(SO(n))+ n, cup(SO(2n)) = cup(SO(2n - 1)) + 1, known to the author thanks to Mamoru Mimura, from a 2008-preprint by Kei Sugata, and the formula (perhaps folkloric) [mathematical expression not reproducible], where [mathematical expression not reproducible] is the highest power of 2 dividing n.

But the problem of finding an explicit formula for cup(SO(n)) was open thus far. The main aim of this note is to solve it by proving that the cup-length of SO( n) can be expressed in the following surprisingly concise way.

Theorem 1.1. For any positive integer n,

cup(SO(n)) = n - 1 + (n - 1)',

where (n - 1)' = [[summation].sup.k.sub.i-1] in [.sub.i][2.sup.i-1] if n - 1 has the dyadic expansion [[summation].sup.k.sub.i-0] in [.sub.i][2.sup.i].

In view of the Elsholz inequality, Theorem 1.1 immediately implies a global lower bound for the Lyusternik-Shnirel'man category of rotation groups.

Corollary 1.1. We have

cat(SO(n)) [greater than or equal to] n - 1 + (n - 1)'.

Due to I. James and W. Singhof [5], N. Iwase, M. Mimura, and T. Nishimoto [3], and N. Iwase, K. Kikuchi, and T. Miyauchi [4], it is known that this lower bound is sharp for n = 1,2,..., 10. Of course, our formula for cup(SO(n)) (Theorem 1.1) is of interest in its own right. But it also enables us to transform the conjecture worded in [4], "this would suggest that cat(SO(n)) = cup(SO(n)) for all n," into the following explicit problem.

Question 1.1. Is it true that cat(SO(n)) = n - 1 + (n - 1)' for n [greater than or equal to] 1?

For odd n (n [greater than or equal to] 3), let q be the unique integer such that [2.sup.q-1] < n < [2.sup.q]. Write [mathematical expression not reproducible] the dyadic expansion of n. Then we have [V.sub.t] < q and Theorem 1.1 yields that cup(SO(n)) < [(n-1)(q+2)/2]. In a similar way, one verifies that cup (SO(n)) < [(n-2)(q+2)/2] for even n. [It is easy to compare these bounds with [n(n-1)/2] = ([??]) = dim(SO(n)).] We thus may state the following weaker (but presumably still very hard) question (whose answer by "No" would of course mean that also Question 1.1 must be answered by "No").

Question 1.2. For a positive integer n, let q denote the unique integer such that [2.sup.q-1] < n < [2.sup.q]. Is it true that cat(SO(n)) < [(n-1)(q+2)/2] for alloddn, n [greater than or equal to] 11, and cat(SO(n)) [less than or equal to] [(n-2)(q+2)/2] for all even n, n [greater than or equal to] 12?

2 Proof of the main result

Poincare duality implies that the cup-length of SO(n) is realized by a cohomology class in the top degree; note that we may identify [[ETA].sup.[n(n-1)/2]] (SO(n)) = [Z.sub.2]. Obviously, if n is odd, then cup (SO(n)) equals the sum of the exponents in

[mathematical expression not reproducible]. (2)

Thus by (2), for odd n, we see that cup(SO(n)) = ([p.sub.1] - 1) + ([p.sub.3] - 1) + *** + ([p.sub.n-4] - 1) + 1; consequently, cup(SO(n + 1)) is obviously the sum of the exponents in the product [mathematical expression not reproducible], since dim(SO(n + 1)) dim(SO(n)) = n (this difference equals the degree in which we have the generator [[beta].sub.n]). Thus indeed, for odd n, cup(SO(n + 1)) = [([p.sub.1] - 1) + ([p.sub.3] - 1) + *** + ([p.sub.n-4] - 1) + 1] + 1 = cup(SO(n)) + 1, as claimed. For even n, a proof is omitted. We have come to the following fact.

Fact 2.1. Let c(n) = cup(SO(n)), n [greater than or equal to] 1, and [[nu].sub.2](n) be the exponent of the highest power of 2 dividing n. Then (i) c(1) = 0; (ii) [mathematical expression not reproducible].

The key observation is the following.

Lemma 2.1. We have c(m + [2.sup.k]) - c(m) = c([2.sup.k] - 1) + [2.sup.k] + 1, if 1 [less than or equal to] m [less than or equal to] [2.sup.k], k [greater than or equal to] 1.

Proof. If m = 1, we have c(1 + [2.sup.k]) - c(1) = c(1 + [2.sup.k]) = c([2.sup.k]) + [2.sup.k] = c([2.sup.k] - 1) + [2.sup.k] + 1 by Fact 2.1 (i) and (ii), and so we assume 1 < m [less than or equal to] [2.sup.k] .By Fact 2.1 (ii), we have [mathematical expression not reproducible], and thus obtain c(m + [2.sup.k]) - c(m) = c(m - 1 + [2.sup.k]) - c(m - 1) = ... = c(1 + [2.sup.k]) - c(1) and is equal to c([2.sup.k] - 1) + [2.sup.k] + 1.

To show the main result, we need the following proposition.

Proposition 2.1. For any k [greater than or equal to] 1, c([2.sup.k] - 1) = k[2.sup.k-1] - 1.

Proof. By Lemma 2.1 with m = [2.sup.k] - 1, we obtain 1 [less than or equal to] m [less than or equal to] [2.sup.k] and c([2.sup.k+1] - 1) = c([2.sup.k] - 1 + [2.sup.k]) = c([2.sup.k] - 1) + c([2.sup.k] - 1) + [2.sup.k] + 1, which yields the following recurrence relation by taking [a.sub.k] = [c([2.sup.k]) + 1/[2.sup.k]]:

[a.sub.k+1] = [a.sub.k] + [1/2], k [greater than or equal to] 1, which is an arithmetic sequence starting with [a.sub.1] = [c([2.sup.1]) + 1/[2.sup.1]] = [1/2], and hence [a.sub.k] = [k/2] and c([2.sup.k] - 1) = [k2.sup.k-1] - 1, k > 1.

Under the above observation, we obtain the main result as follows.

Theorem. We have c(n) = n - 1 + (n - 1)',n [greater than or equal to] 1, where (n - 1)' = [[summation].sup.k.sub.i-1] in [.sub.i][2.sup.i-1]

n - 1 has the dyadic expansion [[summation].sup.k.sub.i-0] in [.sub.i][2.sup.i].

Proof. If n = 1, it is clear by Fact 2.1 (i), and so we assume n [greater than or equal to] 2 and n - 1 [greater than or equal to] 1 has the dyadic expansion [[summation].sup.k.sub.i-0] in [.sub.i][2.sup.i], with [n.sub.k] = 1. We show the formula by induction on k [greater than or equal to] 0.

k = 0: Then n = 2 and c(2) = 1 = 1 + 1' by Fact 2.1 (i) and (ii).

k [greater than or equal to] 1: Let m = n - [2.sup.k], to obtain 0 [less than or equal to] m < [2.sup.k] and c(m) = m - 1 + (m - 1)' by induction hypothesis. Then by Lemma 2.1, we have c(n) = c(m + [2.sup.k]) = c(m) + c([2.sup.k] - 1) + [2.sup.k] + 1 = m - 1 + (m - 1)' + [k2.sup.k-1] + [2.sup.k] = (n - 1) + (n - 1)'. This completes the proof of Theorem.

Acknowledgements. The author thanks Peter Zvengrowski for useful comments related to the presentation of this paper. He is also grateful to the referee for an idea of abbreviating the proof.

References

[1] Borel, A.: Sur la cohomologie des varietes de Stiefel et de certains groupes de Lie, C. R. Acad. Sci. Paris 232 (1951), 1628-1630.

[2] Hatcher, A.: Algebraic Topology, Cambridge University Press, Cambridge, 2002.

[3] Iwase, N.; Mimura, M.; Nishimoto, T.: Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups. Topology Appl. 150 (2005), 111-123.

[4] Iwase, N.; Kikuchi, K.; Miyauchi, T.: On Lusternik-Schnirelmann category of SO(10). Fundamenta Math. 234 (2016), 201-227.

[5] James, I. M.; Singhof, W.: On the category of fibre bundles, Lie groups, and Frobenius maps. Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996), 177-189, Contemp. Math. 227, Amer. Math. Soc., Providence, RI, 1999.

Julius Korbas (*)

Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics, Comenius University Bratislava, Mlynska dolina, SK-842 48 Bratislava 4, Slovakia; e-mail: korbas@fmph.uniba.sk

(*) Part of this research was carried out while the author was a member of two research teams supported in part by the grant agency VEGA (Slovakia). He was also partially affiliated with the Mathematical Institute, Slovak Academy of Sciences, Bratislava.

Received by the editors in October 2016 - In revised form in December 2016.

Communicated by N. Wahl.

Key words and phrases : cup-length; Lyusternik-Shnirel'man category; rotation group (special orthogonal group).
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