Let $M$ be a compact orientable manifold which is homeomorphic to its connected sum with itself $M\# M$. Must $M$ be homeomorphic to a sphere?
I will explain why I am interested (at the risk of being outright foolish or overambitious). The 'equation' $M = M\#M$ is the simplest conceivable equation in the category of compact topological manifolds, just as $x+x=x$ characterises $0$ in an abelian group. Although I am suspicious if $M$ could be a homology sphere, I hope to get a characterisation of spheres.
Thanks for any comments or suggested references.