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Let $M$ be the projective cover (e.g, Gleason1958) of the Cantor set $\{-1,1\}^{\mathbb{N}}$. Let $\textrm{homeo}(M)$ denote the group of all homeomorphisms of $M$. Some of the $\gamma\in\textrm{homeo}(M)$ are extensions of the homeomorphisms of the Cantor set. Some others are the extensions of the homeomorphisms of Polish spaces $K$ for which $M$ is the projective cover of $K$.

Q: Is there a complete explicit description of the homeomorphisms of $M$?

I'd be grateful if you point me to a reference about the properties of $\textrm{homeo}(M)$.

Thanks in advance.

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I claim that every autohomeomorphism of $M$ is conjugate to an extension of an autohomeomorphism of the Cantor space.

Let $B$ be the Stone space of the Cantor space. Let $\overline{B}$ denote its Boolean completion. We shall show that every automorphism of $\overline{B}$ is conjugate to an automorphism of $B$. Recall that $\overline{B}$ is up-to-isomorphism the only complete atomless Boolean algebra with a countable basis (by a basis, I mean a subset $A\subseteq\overline{B}$ where $\overline{B}=\{\bigvee R:R\subseteq A\}$; for example, $B$ is a countable basis for $\overline{B}$). Suppose that $h:\overline{B}\rightarrow \overline{B}$ is an automorphism. Then let $D\subseteq \overline{B}$ be the smallest Boolean subalgebra closed under the operations $h,h^{-1}$ and which contains the countable basis $A$. Then $h$ restricts to an automorphism $g:D\rightarrow D$, and $h$ is the unique automorphism of $\overline{B}$ that extends $g$. Recall that every countable atomless Boolean algebra is isomorphic (the theory of countable atomless Boolean algebras is $\omega$-categorical), so $B$ and $D$ are isomorphic. Let $\phi:B\rightarrow D$ be an isomorphism. Then $\phi$ extends to an automorphism $\Phi:\overline{B}\rightarrow\overline{B}$. In this case, $\Phi^{-1}h\Phi:\overline{B}\rightarrow\overline{B}$ is an automorphism that uniquely extends an automorphism of $B$.

One can extend this argument to conclude that if $(h_n)_{n=0}^\infty$ is a collection of automorphisms of $\overline{B}$, then there is some automorphism $\Phi:\overline{B}\rightarrow\overline{B}$ where for each $j$, the mapping $\Phi^{-1}h_j\Phi$ uniquely extends an automorphism of $B$.

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  • $\begingroup$ +1 Before all, thank you for taking your time to write a detailed answer. I can follow everything but a minor and an important (for me) detail. I'm guessing that $A$ corresponds to a clopen topology basis of the Cantor set, so $A\subseteq B$. Can you open this up? $\endgroup$
    – Onur Oktay
    Commented Aug 23, 2023 at 15:04
  • $\begingroup$ Second, why are $B$ and $D$ isomorphic? This implies on the other side that $M$ contains another homeomorphic copy of the Cantor set, which is invariant under the homeomorphism induced by $h$. $\endgroup$
    – Onur Oktay
    Commented Aug 23, 2023 at 15:05
  • $\begingroup$ I see that you've edited your answer. I didn't know that countable atomless Boolean algebras are isomorphic, a key fact for where I was trying to get. I appreciate your response, thanks again. $\endgroup$
    – Onur Oktay
    Commented Aug 23, 2023 at 18:17

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