Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?



Stable torus that is not a torus
Suppose $M\times S^1$ is homeomorphic to $T^{n+1}$. Then $\pi\_1(M\times S^1) \cong \pi\_1(T^{n+1})$, so $\pi\_1(M)\oplus\mathbb{Z} \cong \mathbb{Z}^{n+1}$, and hence $\pi\_1(M) \cong \mathbb{Z}^n$. Moreover, as $T^{n+1}$ is aspherical, so is $M$. Since $\pi\_1(M) \cong \pi\_1(T^n)$ and both are aspherical, we see that $M$ is homotopy equivalent to $T^n$ (aspherical spaces are uniquely determined up to homotopy by their fundamental group). Finally, by combining the results [here](https://mathoverflow.net/q/109411/21564), a closed manifold of any dimension homotopy equivalent to a torus is in fact homeomorphic to a torus, so $M$ is homeomorphic to $T^n$.


If you ask the same question at the level of diffeomorphism, then I suspect it is not true. As [this answer](https://mathoverflow.net/a/239162/21564) explains, there are exotic tori in every dimension $n \geq 5$. If $M$ is an exotic torus, I don't know if $M\times S^1$ can be diffeomorphic to the standard torus.


**Added Later:** I asked about the diffeomorphism analogue [here](https://mathoverflow.net/q/435116/21564).


