Finite loop space with maximal tori have finite Weyl groups

Abstract

A finite loop space X X is said to have a maximal torus if there is a map f : B T → B X f:BT \to BX where T T is a torus such that rank ⁡ ( T ) = rank ⁡ ( X ) \operatorname {rank} (T) = \operatorname {rank} (X) and the homotopy fibre of f f has the homotopy type of a finite complex. The Weyl group W f {W_f} of f f is the set of homotopy classes w : B T → B T w:BT \to BT such that \[ B T → w B T f ↘ ↙ f B X \begin {array}{*{20}{c}} {BT\xrightarrow {w}BT} \\ {f \searrow \quad \swarrow f} \\ {BX} \\ \end {array} \] homotopy commutes. In this note we prove that W f {W_f} is always finite.