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Suppose the circle group $G = S^1$ acts on $X$.

If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. Note this says something about the space $X^G$ as well as the map $X^G \to X$.

How bad can the fixed-point set be if $X$ is only a closed topological manifold, equipped with a continuous circle action?

  • I'm really looking for a few pathological examples, exhibiting interesting phenomena that can happen.
  • The question concerns both $X^G$ and the map $X^G \to X$.

For added motivation:

Even if $X$ is just a compact Hausdorff space (or even worse, see for example this paper) pullback induces an isomorphism $u^{-1} H_G^*(X) \to u^{-1}H_G^*(X^G)$ of localized modules (here $u$ is the generator of the polynomial algebra $H_G^*(pt)$). However, part of the appeal of the classical fixed-point formula for smooth manifolds is that one can compute explicitly the inverse, using push-forward along $X^G \to X$ and the equivariant Euler to the normal bundle (which gives an equivariant model for a neighborhood of $X^G$). Some obfuscated version of this may hold when $X$ is just a topological manifold, and I'd like to see it play out in pathological examples (if such exist).

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