**Question:** Is there a group $G$ and a CW-complex $X$ such that 


1) $X$ is homotopy equivalent to the circle $S^{1}$.


2) $G$ acts on $X$ 


3) the space of fixed points $X^{G}$ is weakly equivalent to $S^{2}$ ? 



from a circle to higher spheres
A very general answer is given by Tony Elmendorf's paper *Systems of Fixed Point Sets*, <http://www.ams.org/journals/tran/1983-277-01/S0002-9947-1983-0690052-0/>. Very loosely speaking, it says that, if you can write down a reasonable system of fixed sets, it can be realized up to homotopy equivalence.


Take, for example, $G = \mathbb{Z}/2$. Define a contravariant functor $\mathscr{X}$ on the category of orbits of $G$ with $\mathscr{X}(G/e) = S^1$ and $\mathscr{X}(G/G) = S^2$; let the self-map of $G/e$ act on $S^1$ by reflection, with two fixed points, and let the map corresponding to the projection $G/e\to G/G$ map $S^2$ to one of the fixed points in $S^1$. Elmendorf's construction then gives a $G$-CW complex $X$ whose system of fixed sets maps to $\mathscr{X}$ with $X\to \mathscr{X}(G/e)$ and $X^G\to \mathscr{X}(G/G)$ both being homotopy equivalences.


