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An edge of a triangulated manifold 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 triangulated manifold 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?

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"On a $d$-simplex, no edge is contractible. [...] Does every simplicial polytope have a contractible edge?". I think you just answered your own question. –  Dan Petersen Apr 30 '13 at 23:03
    
Thanks for pointing this out! I've added language to indicate I'm interested in nontrivial examples: ie, not the simplex. –  Anand Kulkarni Apr 30 '13 at 23:57
    
It seems that in your first paragraph you don't really talk about general simplicial complexes but only about triangulations of manifolds. –  Fernando Muro May 1 '13 at 9:46
    
I've adjusted this to be more specific. –  Anand Kulkarni May 1 '13 at 16:34
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I'll mention for general interest that Eran Nevo studied edge contractions in his thesis, which is available on the arXiv: arxiv.org/pdf/0709.3265.pdf . He says more about the consequences of an edge contraction than the existence of one, but you might find the algebraic shifting arguments in there interesting. –  Russ Woodroofe May 2 '13 at 17:57

1 Answer 1

I brought this thread to the attention of Eran Nevo, who gave some more references, and pointed out that the answer to the converse of your question (2) is "no".

In his thesis, which I gave the arXiv link to above, and in the paper version of this "Higher minors and Van Kampen's obstruction", Nevo defines a strongly edge decomposable sphere to be a sphere which can be reduced to the simplex by edge contractions. He shows that in any PL-manifold (in any dimension), an edge is contractible if and only if it satisfies the link condition.

Then Satoshi Murai in "Algebraic shifting of strongly edge decomposable spheres" shows that any squeezed sphere is strongly edge decomposable. (Murai also has an earlier paper which is related, "Generic initial ideals and squeezed spheres".) Since there are more squeezed $d$-spheres than polytopes for $d \geq 5$, there are non-polytopal spheres where an edge contraction leaves a polytopal sphere. Though I didn't see $d=4$ settled explicitly one way or the other in the papers I looked at, I suspect that there are non-polytopal squeezed 4-spheres as well (and probably this is known). Every squeezed $3$-sphere is polytopal.

Another paper which is somewhat relevant is Babson and Nevo "Lefschetz properties and basic constructions on simplicial spheres".

I also thought a bit about the forward direction of your question (2). If $vw$ is a contractible edge in the boundary of the convex hull of $V$, it looks plausible to me that the edge contraction should correspond to passing to (boundary of) the convex hull of $V \setminus \lbrace v,w \rbrace \cup x$, where $x$ is a point on $vw$. For as you 'slide' $v$ and $w$ together, the link condition seems to prevent new faces from being created. (Obviously, there is a lot to be checked with this idea!)

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