Question

I ask this question because in the process of reviewing for my topology comp, I began rereading Alg Topology by Hatcher. In the introduction is the famous Bing's House of Two Rooms. I thought this was an interesting example and began reading about it on the web (procrastinating). Several sites note that Bing's house is contractable (as described in Hatcher) but not collapsible. The definition of collapsible does not appear in any of my topology or alg topology books (Munkres, Hatcher, Spanier) and the only definition I have found is on wikipedia. So this brings me to my question, is collapsible a useful topological concept? And can anyone show me why Bing's house is not collapsible (I guess I probably do not fully comprehend the definition)?

Answer 1

There is a simple reason for appreciating collapsible objects: a collapsible (PL) n-manifold is always (PL) homeomorphic to a disc! (Although a contractible one may not, for instance in dimension 4.) For a proof, see [Rourke C.P., Sanderson B.J. Introduction to piecewise-linear topology (Springer, 1972)].

Concerning Bing's house, you cannot do any elementary collapse: in order to do such a collapse, some point must have a link (PL)-homeomorphic to a disc (of some dimension). But the points in Bing's house have links homeomorphic to a circle, a circle with a radius, or a Mercedes Benz symbol.

Answer 2

The notion of an elementary collapse is the key construction in the definition of a simple homotopy-equivalence. Simple homotopy type is a refinement of homotopy type, and it has many uses.

One example: The s-cobordism theorem is the main structure theorem for high-dimensional manifolds, and it has to do with the question of when two simple-homotopy equivalent manifolds (technically s-cobordant manifolds) are diffeomorphic.

You might want to take a look at Marshall Cohen's introduction to simple homotopy theory textbook. One of the key examples there is that simple homotopy type is the same as homeomorphism type for lens spaces, but homotopy type is a different, weaker relation.

Whitehead torsion and Reidemeister torsion are simple homotopy invariants. In a sense you can think of simple homotopy theory as a space-level analogue of elementary row and column operations on a chain complex. So it gives you a sense for why it should somehow be more relevant to forming a bridge between topology and algebraic constructions.

As for Bing's house, I think the answer is kind of simple. Where would you start the collapse?

Answer 3

To add a little to Ryan's answer:

This topic is maybe not exactly a part of algebraic topology. It's more like an area of application of algebraic topology to certain important special classes of spaces.

When $A$ is a subcomplex of $B$ (both of them finite) and $B$ collapses to $A$, then the inclusion map $A\to B$ is said to be a simple homotopy equivalence, as is any left inverse of such an inclusion, and more generally any map between finite complexes that is homotopic to a composition of such things. A homotopy equivalence $A\to B$ between finite complexes determines an element of the Whitehead group $Wh(G)$ of the fundamental group $G=\pi_1(A)$ (a certain abelian group that depends functorially on $G$ -- the quick definition is take the direct limit of $GL_n(Z[G])$ as $n$ goes to infinity, abelianize, and kill the the invertible $1\times 1$ matrices $g\in G$ and $-1$). It is simple if and only if this element is zero. (Sometimes the latter is taken as definition of simple.) The group $Wh(1)$ is trivial, so every homotopy equivalence between simply-connected finite complexes is simple. The house with two rooms shows that the inclusion of a subcomplex can be simple even if there is no collapse; there is a larger complex collapsing both to the house and to the point.

The question of whether a homotopy equivalence is simple is unchanged by subdivision of a complex, so sometimes you can prove that a given homotopy equivalence is not homotopic to any simplicial or cellular isomorphism by proving that it has nontrivial Whitehead torsion. If you can enumerate all the (homotopy classes of) homotopy equivalences from $A$ to $B$ and you find that all of them have nontrivial torsion, then the spaces are really different. Eventually the topological invariance of Whitehead torsion was proved, making for stronger statements.

$Wh(G)$ is trivial for lots of (potentially for all) torsion-free groups $G$, but is usually nontrivial for finite $G$.

Answer 4

If you're willing for "useful" to apply to a field other than topology itself: collapses are quite important in combinatorics. Elementary collapses correspond (via Euler characteristics) to matching up equi-numerous objects counted with opposite signs in inclusion-exclusion type problems.

Discrete Morse theory is the generalized version of this. The basic idea in discrete Morse theory is that one can collapse faces of adjacent dimension in skeleta of a simplicial complex, then glue on higher dimensional faces in a way respecting the collapsing (without changing homotopy type). Discrete Morse theory has been a quite important tool in topological combinatorics for the past 10 or 15 years. See the papers of Forman: "Morse theory for cell complexes" introduced the topic (though I should mention that the basic idea was discovered by Ken Brown in "The geometry of rewriting systems: a proof of the Anick-Groves-Squier Theorem"), or "Topics in combinatorial differential topology and geometry" is a survey article.

Discrete Morse theory can also be seen as a generalization of the theory of shellings (also based on a collapsing idea), which has been important in topological and algebraic combinatorics since the late 70s/early 80s.