View source for Collapsibility

''of a triangulation''

Originally, a property of simplicial complexes (cf. [[Simplicial complex|Simplicial complex]]), extending naturally to simplicial sets (cf. [[Simplicial set|Simplicial set]]) and cell complexes (cf. [[Cell complex|Cell complex]]). For a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103101.png" /> of polyhedra or complexes (see [[Polyhedron, abstract|Polyhedron, abstract]]), one says there is an elementary collapse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103103.png" />, and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103104.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103105.png" /> is the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103106.png" /> with precisely two more simplexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103108.png" />, one a maximal face of the other. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c1103109.png" /> collapses on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031010.png" /> if there is a sequence, possibly transfinite, of elementary collapses

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031011.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031012.png" /> for a limit ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031013.png" /> must be the intersection of the preceding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031014.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031015.png" /> is called a spine of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031016.png" />, and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031018.png" /> collapses to a point, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031019.png" /> is collapsible and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031020.png" />.

Collapses were first introduced [[#References|[a18]]] for two applications which are fundamental for [[Piecewise-linear topology|piecewise-linear topology]]: description of the internal structure of a complex, and the definition of simple homotopy type. The  "modern" , functorial, definition is (see [[Simple homotopy type|Simple homotopy type]]) that two complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031022.png" /> have the same simple homotopy type if there exists a homotopy equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031023.png" /> with zero [[Whitehead torsion|Whitehead torsion]]. This is equivalent to Whitehead's constructive definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031025.png" /> can be joined by a  "zigzag"  chain of collapses

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031026.png" /></td> </tr></table>

(for finite complexes; it can be extended to infinite complexes). It was clear at once that collapsible complexes (and the underlying (piecewise-linear) polyhedra and topological polyhedra) are an interesting class of spaces: the best behaved contractible polyhedra. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031027.png" />-dimensional contractible complex (topological tree) is collapsible. However, there are simple examples of contractible but not collapsible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031028.png" />-dimensional complexes, such as the house with two rooms [[#References|[a4]]] and the dunce hat [[#References|[a19]]].

The centre of interest in collapsible polyhedra is the Zeeman conjecture, stating that for every contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031029.png" />-dimensional complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031031.png" /> is collapsible. This was proposed by E.C. Zeeman [[#References|[a19]]], who showed that it implies the [[Poincaré conjecture|Poincaré conjecture]]. Nearly twenty years later, D. Gillman and D. Rolfsen showed [[#References|[a7]]] that the conjectures are equivalent, at least if the Zeeman conjecture is somewhat restricted, as explained below. This equivalence has the happy effect of  "blowing-up"  the problem into something that has parts, unlike the original Poincaré conjecture, which merely says that certain spaces cannot exist. Indeed, another part of Zeeman's conjecture has been shown [[#References|[a13]]] to be equivalent to a conjecture of J.J. Andrews and M.L. Curtis [[#References|[a1]]] on contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031032.png" />-dimensional complexes, which acquires much greater interest because of its equivalence in turn with a major conjecture in combinatorial group theory. These things are explained below.

Note that there is another field in which collapsibility is prominent, and which is less dependent upon conjecture. It revolves around the question: What are the topological types of injective or hyperconvex metric spaces? (See [[Metric space|Metric space]].) N. Aronszajn and P. Panitchpakdi speculated [[#References|[a3]]] that they might be just the (metrizable) AR's (see [[Retract of a topological space|Retract of a topological space]]); and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031033.png" />-dimensional spaces it is true [[#References|[a16]]]. However, J. Isbell showed [[#References|[a10]]] that a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031034.png" />-dimensional polyhedron underlies an injective metric space if and only if it underlies a collapsible complex, and observed that this might extend to higher dimensions. At least half of it does, in that finite collapsible complexes are homeomorphic with injective metric spaces [[#References|[a11]]].

Why is the Zeeman conjecture restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031035.png" />-dimensional contractible complexes? Zeeman did not say, but soon there was speculation about an extended Zeeman conjecture in high dimensions. M. Cohen showed in 1977 [[#References|[a6]]] that that is false beyond dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031036.png" />.

Another point around the boundary of the Zeeman conjecture is as follows. Of course, contractibility is a topological property, while collapsibility is not. Nevertheless, it may well be true that for every [[Triangulation|triangulation]] of a contractible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031037.png" />-dimensional polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031039.png" /> is collapsible (the full Zeeman conjecture). In this connection it is easy to show that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031040.png" />-dimensional polyhedra, collapsibility is a topological invariant. In fact, for compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031041.png" />-dimensional polyhedra it is equivalent [[#References|[a10]]] to topological collapsibility (defined below). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031042.png" />-dimensional polyhedra, even the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031043.png" />-dimensional ball, collapsibility depends on the triangulation [[#References|[a8]]].

Here is a very rough sketch of Zeeman's proof that his conjecture implies Poincaré's. Given a fake <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031045.png" />-dimensional sphere (a simply connected closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031046.png" />-dimensional manifold) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031047.png" />, remove an open piecewise-linear 3-dimensional ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031048.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031049.png" /> collapses to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031050.png" />-dimensional complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031051.png" /> which is easily shown to be contractible. So, given the Zeeman conjecture, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031052.png" />. By a classical theorem of J.H.C. Whitehead (see [[#References|[a17]]]) this makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031053.png" /> a piecewise-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031054.png" />-dimensional ball, and the rest is easy.

It is known that the Poincaré conjecture is equivalent to the Zeeman conjecture restricted to special spines (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031055.png" />-dimensional manifolds with boundary). Special (also standard) polyhedra are compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031056.png" />-dimensional polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031057.png" /> having a piecewise-linear CW-decomposition whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031058.png" />-skeletons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031059.png" /> are topologically defined as follows. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031060.png" /> is the set of points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031061.png" /> is locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031062.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031063.png" /> is the set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031064.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031065.png" /> is locally three half-planes with a common edge; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031066.png" /> is the remainder, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031067.png" /> must be locally four trihedral angles sharing six face angles in pairs. Every connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031068.png" />-dimensional manifold with boundary has a special spine [[#References|[a5]]].

The original Andrews–Curtis conjecture [[#References|[a1]]] concerns balanced presentations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031069.png" />-element group, i.e., presentations by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031070.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031072.png" /> relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031073.png" />. It states that every such presentation can be reduced to the trivial presentation in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031074.png" /> by transformations of four types, three from J. Nielsen's 1919 paper [[#References|[a14]]] and, fourth, replacing a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031075.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031076.png" /> for any word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031077.png" />. (See the expository paper [[#References|[a2]]].) A number of equivalent propositions are known [[#References|[a9]]]. The basic topological one is that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031078.png" />-dimensional contractible complex can be joined to a point by a zigzag chain of collapses, all within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031079.png" />-dimensional complexes. (If one replaces  "2"  and  "3"  by  "n"  and ` "n+ 1" , with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031080.png" />, this is a theorem of Whitehead [[#References|[a19]]].) As for the Zeeman conjecture, its specialization to contractible special polyhedra which are not imbeddable in any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031081.png" />-dimensional manifold is equivalent to the Andrews–Curtis conjecture [[#References|[a13]]].

Topological collapsibility [[#References|[a15]]] is the existence of a free contraction to a point, i.e., a [[Homotopy|homotopy]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031082.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031084.png" />, and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031085.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031086.png" />. For compact polyhedra, collapsibility implies injectivity [[#References|[a11]]] and injectivity implies topological collapsibility [[#References|[a10]]]. There seems to be no known example (1996) of a topologically collapsible polyhedron which has no collapsible triangulation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.J. Andrews,   M.L. Curtis,   "Free groups and handlebodies"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 192–195</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.J. Andrews,   M.L. Curtis,   "Extended Nielsen operations in free groups"  ''Amer. Math. Monthly'' , '''73'''  (1966)  pp. 21–28</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Aronszajn,   P. Panitchpakdi,   "Extension of uniformly continuous transformations and hyperconvex metric spaces"  ''Pacific J. Math.'' , '''6'''  (1956)  pp. 405–439</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Bing,   "Some aspects of the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031087.png" />-manifolds related to the Poincaré conjecture" , ''Lectures on Modern Mathematics'' , '''II''' , Wiley  (1964)  pp. 93–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.G. Casler,   "An imbedding theorem for connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031088.png" />-manifolds with boundary"  ''Proc. Amer. Math. Soc.'' , '''16'''  (1965)  pp. 559–566</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.M. Cohen,   "Whitehead torsion, group extensions, and Zeeman's conjecture in high dimensions"  ''Topology'' , '''16'''  (1977)  pp. 79–88</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Gillman,   D. Rolfsen,   "The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture"  ''Topology'' , '''22'''  (1983)  pp. 315–323</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.E. Goodrick,   "Non-simplicially collapsible triangulations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031089.png" />"  ''Proc. Cambridge Philos. Soc.'' , '''64'''  (1968)  pp. 31–36</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  "Two-dimensional homotopy and combinatorial group theory"  C. Hog-Angeloni (ed.)  W. Metzler (ed.)  A. Sieradski (ed.) , ''Lecture Notes'' , '''197''' , Cambridge Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J.R. Isbell,   "Six theorems about injective metric spaces"  ''Comment. Math. Helv.'' , '''39'''  (1964)  pp. 65–76</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J. Mai,   Y. Tang,   "An injective metrization for collapsible polyhedra"  ''Proc. Amer. Math. Soc.'' , '''88'''  (1983)  pp. 333–337</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Mai,   Y. Tang,   "On the injective metrization for infinite collapsible polyhedra"  ''Acta Math. Sinica (N.S.)'' , '''7'''  (1991)  pp. 13–18</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S.V. Matveev,   "The Zeeman conjecture for non-thickenable special polyhedra is equivalent to the Andrews–Curtis conjecture"  ''Siberian Math. J.'' , '''28'''  (1987)  pp. 917–928  ''Sibirsk. Mat. Zh.'' , '''28'''  (1987)  pp. 66–80;218</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J. Nielsen,   "Über die Isomorphismen unendlicher Gruppen ohne Relation"  ''Math. Ann.'' , '''79'''  (1919)  pp. 269–272</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R. Piergallini,   "Topological collapsing and cylindrical neighborhoods in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031090.png" />-manifolds"  ''Topology Appl.'' , '''20'''  (1985)  pp. 257–278</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  R.L. Plunkett,   "Concerning two types of convexity for metrics"  ''Arch. Math.'' , '''10'''  (1959)  pp. 42–45</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  C.P. Rourke,   B.J. Sanderson,   "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  J.H.C. Whitehead,   "Simplicial spaces, nuclei and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110310/c11031091.png" />-groups"  ''Proc. London Math. Soc.'' , '''45'''  (1939)  pp. 243–327</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  E.C. Zeeman,   "On the dunce's hat"  ''Topology'' , '''2'''  (1964)  pp. 341–358</TD></TR></table>