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Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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Is the fixed point set of a self-diffeomorphism of odd order orientable?

I believe this is true. Lemma For $\mathbb{Z}_p$, $p$ odd, all nontrivial irreducible real representations are 2-dimensional. (They really come from a complex representation, forgetting the complex …
Eric Schlarmann's user avatar