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.



A question on connected sum of compact manifolds
Let us suppose that $dim(M)\geq 3$ then we have that:


* $\pi\_1(M \# M)\cong \pi\_1(M)\*\pi\_1(M)$,
* $H\_\*(M;\mathbb{Z})\cong H\_\*(M;\mathbb{Z})\oplus H\_\*(M;\mathbb{Z})$ when $\*< dim(M)$.


As $M$ is compact this implies that $\pi\_1(M)$ is finitely presented thus that $M$ is simply connected (\*). Together with the second point it implies that $M$ has the homotopy type of a sphere. And you conclude using the Poincaré conjecture that $M$ is homeomorphic to a sphere.


Edit (\*): we have to use a non-trivial theorem in group theory about the minimal number of generators $m(G)$ of a group $G$. Namely we have $m(G\*H)=m(G)+ m(H)$ (conjectured by Levi and proved independently by Gruschko and Neumann).


