If $X \times X \simeq \mathbb{R}^{2}$, then $X \simeq \mathbb{R}$? I came across this question few years ago, but I have not yet reached a satisfactory answer.
If a product of a topological space $X$ with itself is homeomorphic to the real plane $\mathbb{R}^{2}$, must $X$ be homeomorphic to the real line $\mathbb{R}$? Here I am not assuming a priori that $X$ is a manifold.

Note that I am not an expert in this, but I think this paper is what you are looking for:
Non-manifold factors of Euclidean spaces, Fund. Math. 68 (1970), 159–177, MR275396 describes classes $\Gamma_m$ and $\Gamma_n$ where if $X \in \Gamma_m$ and $Y \in \Gamma_n$, then $X \times Y \cong \mathbb{R}^{n+m}$.
