Is there a principal bundle $P\to X$ such that $P$ has the fixed point property but $X$ does not have? Is there an example of this situation where $P,X$ and the fibers are compact smooth manifolds?(In particular the group is a compact Lie group)?
2026-03-25 06:34:22.1774420462
Fixed point property for the total space and base space of a principal bundle
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The group action on $P$ implies no principal bundle can have the fixed point property, unless the fiber (group) $G$ is trivial. If $G$ is not trivial, note that the morphism (group action) induced on $P$ by any non-zero element in $G$ lacks a fixed point. If $P$, a principal bundle over $X$ has the fixed point property, then $P=X$ . Therefore $X$ has the fixed point property.