An edge of a simplicial complex is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is noncontractible.

Not every edge of a simplicial complex is contractible. For example, on a triangular bipyramid, the edges gluing the two pyramids together are noncontractible, while all other edges are contractible (yielding a simplex). On a $d$-simplex, no edge is contractible.

Dey, Edelsbrunner, Guha, and Nekyahev provide exact conditions for when an edge of a 2- or 3-manifold is contractible, noting that an edge $ab$ between vertices $a$ and $b$ is contractible iff $link(ab) = link(a) \cap link(b)$. I have found little in the literature about conditions for the contractibility of edges on higher-dimensional objects.

My questions concern the existence of contractible edges on polytopal $d$-spheres, ie, simplicial polytopes.

1. Does every simplicial polytope other than the simplex have a contractible edge?

2. Does contraction of an edge of a simplicial polytope always yield another polytope?

3. Are these questions any easier to answer if we restrict the question to 4-polytopes?