# Hamiltonicity for triangulations of the 3-sphere

A classical theorem of Whitney states that the 1-skeleton of every triangulation of the 2-sphere $$\mathbb{S}^2$$ has a Hamilton cycle as long as each of its 3-cycles bounds a triangle.

I'm wondering if this extends to higher dimensions:

Prove that for every triangulation of $$\mathbb{S}^3$$, the 2-skeleton contains a 'Hamilton sphere', i.e. a homeomorph of $$\mathbb{S}^2$$ containing all vertices, unless it contains certain substructures (which are for you to find).