What is known about $\operatorname{Aut}(\mathbb{I}^n)$ A few months ago, I asked a related question: Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$? It was interesting for me to know that $\operatorname{Aut}(\mathbb{I})\,$ has a normal subgroup of index $2$. Now what other things are known about $\operatorname{Aut}(\mathbb{I}^n)$?
 I guess they are all non-isomorphic . Is there an easy way to show this like the answer in the question which I put a link for?

Another thing: I never studied geometric topology before. Are the above questions closer to geometric topology than algebraic topology? 

The reason why I made this guess is because I saw this page.

I am also trying to get an idea of what Geometric topology is about.

Thank you.
You can generalise the result that $Aut(I)$ has an order 2 subgroup.

Lemma 1 Let $Q_d$ be the edge-graph of a $d$-dimensional hypercube. The automorphism group of $Q_d$ is isomorphic to a wreath product $\Sigma_2\wr\Sigma_d$, also called the hyperoctohedral group in dimension $d$. Here $\Sigma_i$ is the symmetric group on $i$ elements.

Corollary 2 $$|Aut(Q_d)|=2^dd!$$

Theorem 3 The automorphism group $Aut(I^d)$ has a subgroup of order $2^dd!$.

The above is a consequence of corollary 2 and $Aut(Q_d)$ being realised as a subgroup of $Aut(I^d)$. A proof of the above lemma can be found here[1].

[1] Frank Harary. The Automorphism Group of a Hypercube. J. UCS 6(1): 136-138 (2000).
You can generalise the result that $Aut(I)$ has an order 2 subgroup.

Lemma 1 Let $Q_d$ be the edge-graph of a $d$-dimensional hypercube. The automorphism group of $Q_d$ is isomorphic to a wreath product $\Sigma_2\wr\Sigma_d$, also called the hyperoctohedral group in dimension $d$. Here $\Sigma_i$ is the symmetric group on $i$ elements.

Corollary 2 $$|Aut(Q_d)|=2^dd!$$

Theorem 3 The automorphism group $Aut(I^d)$ has a subgroup of order $2^dd!$.

The above is a consequence of corollary 2 and $Aut(Q_d)$ being realised as a subgroup of $Aut(I^d)$. A proof of the above lemma can be found here[1].

[1] Frank Harary. The Automorphism Group of a Hypercube. J. UCS 6(1): 136-138 (2000).