# Diffeomorphisms between factors in diffeomorphic product manifolds

Let $M$, $N$ and $P$ be three smooth manifolds such that $M \times N$ is diffeomorphic to $M \times P$. I need to know about some conditions under which one can deduce that $N$ is diffeomorphic to $P$. For example is it sufficient that $N$ and $P$ to be homotopy equivalent?

Thanks

• Not even homeomorphism is enough, as evinced by the existence of several smooth structures on $S^7$. – Mariano Suárez-Álvarez Apr 14 '13 at 8:55

Take two exotic $\mathbb{R}^4$ (see Wikipedia's exotic $\mathbb{R}^4$ article) as $P$ and $N$, i.e. $P$ and $N$ are homeomorphic to $\mathbb{R}^4$ but $P$ is not diffeomorphic to $N$. Take $M = \mathbb{R}^n$ with $n>0$, then $M\times N$ and $M\times P$ are homeorphic to $\mathbb{R}^{n+4}$. Since there is only one diffeomorphism type for $\mathbb{R}^{n+4}$, we conclude that $M\times N$ is diffeomorphic to $M\times P$.
If $M$ is a $T_1$-space (for example Hausdorff) and $M\times N$ is homeomorphic to $M\times P$, then $N$ is homeomorphic to $P$.
The condition of $M$ being $T_1$ is to ensure that for $m_0\in M$ fixed, the inclusion $N\to \{m_0\}\times N$ and the projection $\{m_0\}\times P\to P$ are homemorphisms.