 Manifold criterion for a suspension

For a nonempty compact topological manifold $X^d$ without boundary, its suspension is a topological manifold without boundary exactly when $X$ is homeomorphic to $S^d$. The forward implication is a local homology from a link calculation at a suspension vertex: ${\tilde{H}}_j(X)$ is $\mathbb{Z}$ in degree $d$ and zero elsewhere. For $d\ge 2$, put a coordinate ball between two nested cone neighbourhoods. The inclusion of their punctures is a homotopy equivalence but factors through the simply connected punctured ball, forcing ${\pi }_1(X)=0$. Thus $X$ is a homotopy sphere, and the topological generalized Poincare theorem gives the result. The dimensions zero and one follow from the finite-set and circle classifications. Conversely $S(S^d)\cong S^{d+1}$. If manifolds with boundary are allowed, the interval and its disk suspension show why this formulation needs the boundary restriction.