A split action associated with a compact transformation group

Schwartzman, Sol

doi:10.1090/S0002-9939-1981-0630061-4

A split action associated with a compact transformation group
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Proc. Amer. Math. Soc. 83 (1981), 817-824 Request permission

Abstract:

We associate with an effective action of a compact connected Lie group as a pathwise connected space $X$ a split action of a quotient group $G/K$ on the quotient space $X/K$. One application of the main theorem states that if $X$ is a compact oriented manifold whose principal cohomology class is a cup product of one-dimensional classes then the action of $G$ on $X$ splits. We prove this in the differentiate case; the topological case has since been dealt with by Schultz.

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