L-S categories of simply-connected compact simple Lie groups of low rank

Abstract

We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal $\widebarΔ_5$ is given by ${\nu }^2$, using a Toda bracket and a generalised cohomology theory $h^{\ast }$ given by $h^{\ast }(X,A)=\{X/A,\mathbb{S}[0,2]\}$, where $\mathbb{S}[0,2]$ is the 3-stage Postnikov piece of the sphere spectrum $\mathbb{S}$. This method also yields a general result that $\cat(Sp(n)) \geq n+2$ for $n\ge 3$, which improves the result of Singhof.

12 pages