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View a plane tree drawn in $\mathbb{R}^2$ as a joining of geometric (straight) segments at endpoints such that (a) they avoid intersecting one another (except where they share a vertex), and (b) they avoid creating a cycle, which would enclose a positive planar area. I am interested in the generalization to $\mathbb{R}^3$ as follows. Join together (flat) polygons, glued edge-to-edge, such that (a) they avoid intersecting one another (except where they share vertices and/or edges), and (b) they avoid enclosing (water-tightly) a positive volume.

My question is:

Is there a name for this construct? Has it been studied?

I am mainly seeking references to any literature on this or related concepts. Of course there is a generalization to $\mathbb{R}^d$, but I would be happy to learn of work just generalizing plane trees to ??? in $\mathbb{R}^3$. I cannot think of what it might be named: open panel structures? It's come up in my work, and I would be delighted to christen it, but surely it has been studied...?

Thanks in advance!

Addendum. As Greg Kuperberg kindly explained, the concept I described is a collapsible complex. It is usually defined for simplicial complexes, but works as well when the constituents are polytopes rather than simplices, e.g., polygons in $\mathbb{R}^3$.

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2 Answers 2

up vote 9 down vote accepted

The standard name for such a subset of $\mathbb{R}^3$ is a contractible (compact) polyhedron if, in addition to your criteria, you also demand that it be simply connected. Of course you can think about contractible polyhedra of any dimension, not just 1 or 2, and embedded in any Euclidean space, or not embedded at all.

But, after choosing that name, there is a big surprise that was discovered by Bing and Borsuk. Namely, that a contractible 2-dimensional polyhedron doesn't have to have a free edge! Bing's version of it is called the house with two rooms, and it is the object on the left:

alt text

(The other part of the diagram shows how you can fill the house with bricks to prove that a ball collapses onto it.)

To get something more tree-like, you need to impose more restrictive conditions. Associated with the examples of Bing and Borsuk, a polyhedron is called collapsible if it is either a disk, or a simpler collapsible polyhedron with a disk attached along some embedded edge. Collapsible polyhedra are one type of more strictly tree-like polyhedra, and I guess that they have been studied since they have a name.

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@Greg: Oh, this is very useful! Contractible, and especially collapsable polyhedra. Thanks so much! Already investigating collapsable polyhedra I've encountered the Zeeman Conjecture, which may now(?) be established by the proof of the Poincaré Conjecture. –  Joseph O'Rourke Mar 19 '11 at 23:32
    
NB, another notion of a tree-like polyhedron is one which is CAT(0) with Euclidean facets. By definition, such a polyhedron comes with a metric. Without its metric, such a polyhedron is collapsible, I think, but I would suppose that not every collapsible polyhedron has a CAT(0) structure. –  Greg Kuperberg Mar 19 '11 at 23:41
    
@Joseph Only part of the Zeeman conjecture is equivalent to the Poincare conjecture. Another part is equivalent to the Andrews-Curtis conjecture, which is still open. eom.springer.de/c/c110310.htm –  Greg Kuperberg Mar 19 '11 at 23:57
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Also check out the House with One Room notes form Allen Hatcher

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