L-S Categories of Simply-connected Compact Simple Lie Groups of Low Rank

Abstract

Abstract. We determine the L-S category of Sp (3) by showing that the 5-fold reduced diagonal ∆ 5 is given by ν 2 , using a Toda bracket and a generalised cohomology theory h * given by h * (X, A) = {X/A, S[0, 2]}, where S[0, 2] is the 3-stage Postnikov piece of the sphere spectrum S. This method also yields a general result that cat(Sp(n)) ≥ n + 2 for n ≥ 3, which improves the result of Singhof [17].

L-S category of quaternionic Stiefel manifolds

“…By setting n − k = 0 in Theorem 1, one gets the following result of Iwase and Mimura [3]. Corollary 3. cat(Sp(n)) n + 2 for n 4.…”

“…The purpose of this paper is to give lower bounds for the L-S category of quaternionic Stiefel manifolds X n,k . We will employ the same cohomology theory h * as which was used by Iwase and Mimura [3] to determine cat(Sp (3)). We will investigate the Atiyah-Hirzebruch spectral sequence for h * and calculate the ring h * (X n,n−k ).…”

Maps of degree 1 and Lusternik–Schnirelmann category

“…The group G 2 is 2-connected and cat G 2 = 4, [11]. Now the result follows from Theorem 5.1 with N = G 2 and q = 3 because 14 = dim G 2 ≤ 2q cat G 2 − 4 = 20.…”

Lusternik-Schnirelmann category of 𝐒𝐩𝐢𝐧(9)

“…In [16], Iwase-Mimura-Nishimoto gave such a refinement in the case when the base space B is non-simply connected. But in this paper, we give another refinement in the case when the fibre bundle is a principal bundle over a double suspension space: Let G be a compact Lie group with a cone-decomposition of length m:…”

“…Here we give its definition in a slightly general fashion, which is inspired by the proof of cat(Sp(2)) = 3 given in [14].…”