Finite loop space with maximal tori have finite Weyl groups

Smith, Larry

doi:10.1090/S0002-9939-1993-1181174-0

Finite loop space with maximal tori have finite Weyl groups
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Proc. Amer. Math. Soc. 119 (1993), 299-302 Request permission

Abstract:

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

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