# The Stone-Čech compactification of the fixed point set

Let $$G$$ be a discrete group and $$X$$ be a Tychonoff $$G$$-space. Then there exists a $$G$$-action on Stone-Čech compactification $$\beta X$$. If the fixed point set $$X^{G}\neq \emptyset$$, then the Stone-Čech compactification of the fixed point set is the fixed point set of the Stone-Čech compactification, i.e., $$\beta \left( X^{G}\right) =\left( \beta X\right) ^{G}$$?

• $\beta$ is a left adjoint functor, and taking $G$-fixed points is a limit. So I suspect this isn't true as in general left adjoint don't preserve limits. – Niall Oct 23 '18 at 10:12
• To start with, one has a canonical continuous map $\beta(X^G)\to (\beta X)^G$. Also the assumption $X^G\neq\emptyset$ is irrelevant as it can be artificially enforced by adding an isolated fixed point to $X$. – YCor Oct 23 '18 at 22:51

Not without further assumptions. First create an ordered space $$X$$ by identifying $$\langle0,\omega_1\rangle$$ and $$\langle1,\omega_1\rangle$$ in the product $$2\times(\omega_1+1)$$ to a point, $$\Omega$$ say. Then make the product $$X\times X$$ and take out $$\langle\Omega,\Omega\rangle$$. You can rotate the resulting space $$Y$$ around that hole over $$\pi$$ by mapping $$\bigl<\langle i,\alpha\rangle,\langle j,\beta\rangle\bigr>$$ to $$\bigl<\langle 1-i,\alpha\rangle,\langle 1-j,\beta\rangle\bigr>$$ and similarly for points with $$\Omega$$ as a coordinate. This map $$f:Y\to Y$$ has no fixed points but $$\beta Y=X$$ and $$\beta f$$ does have a fixed point: $$\langle\Omega,\Omega\rangle$$.
Now take the sum $$Y\cup\{0\}$$ and extend $$f$$ by $$f(0)=0$$. Then $$f$$ has one fixed point and $$\beta f$$ has two. As $$f$$ is its own inverse this gives us an action of $$\mathbb{Z}/2\mathbb{Z}$$ that is a counterexample.