An Uncountable Group of Fibre Homotopy Equivalences

Abstract. Let $G $ be a compact connected Lie group which is not a torus and let $T$ be a maximal torus of $G $. For the fibre bundle with structure group $G$ $G/Tarrow BTarrow BG $, one can consider the group of fibre homotopy equivalences of this fibration, which we denote by $C(BT) $. In this note we show that $\mathcal{L}(BT) $ is a semi-direct product of a rational vector space and the Weyl group $W(G) $ with the operator induced by the natural action $W(G)\cross BTarrow BT $. 1