It we have that $n$-manifolds $M$, $M'$, $N$, and $N'$ are compact and orientable, does it follow that if $M\#N$ is homeomorphic to $M'\#N'$, then $M$ is homeomorphic to $M'$ or $N'$ and $N$ is homeomorphic to $N'$ or $M'$?
2026-04-02 00:31:18.1775089878
Connected sum of compact and orientable manifolds
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No - take, for example, $M$ to be the connected sum of $3$ $2$-tori, $N$ a torus, and $M'=N'$ both the connected sum of $2$ tori.
Both $M \# N$ and $M' \# N'$ give the compact orientable manifold of genus $4$.