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Finite loop space with maximal tori have finite Weyl groups


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 119 (1993), 299-302
MSC: Primary 55P35
DOI: https://doi.org/10.1090/S0002-9939-1993-1181174-0
MathSciNet review: 1181174
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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{displaymath}\begin{array}{*{20}{c}} {BT\xrightarrow{w}BT} \\ {f \searrow \quad \swarrow f} \\ {BX} \\ \end{array} \end{displaymath}

homotopy commutes. In this note we prove that $ {W_f}$ is always finite.

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DOI: https://doi.org/10.1090/S0002-9939-1993-1181174-0
Article copyright: © Copyright 1993 American Mathematical Society

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