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)?
[Math] Is the topological concept of collapsible useful
at.algebraic-topologygt.geometric-topology
Related Solutions
Here is a statement that goes into the direction you are looking for:
When $X$ and $Y$ are smooth varieties over $\mathbb C$, then $f$ is flat if and only if every fiber of $f$ has dimension $\dim X - \dim Y$.
Thiis is perhaps not quite as well-known as it should be; I learned it as a student in my algebraic geometry class taught by Jens Franke. I would also be glad if someone could tell me a reference, as Jens Franke doesn't seem to be planning to turn his lecture notes into a book...
(Here is an example of one of the several precise statements he proved: Let $f \colon X \to Y$ be a morphism of finite type between locally Noetherian prescheme, such that $X$ is Cohen-Macaulay and $Y$ is regular. Then $f$ is flat if either of the following two conditions holds:
- For every irreducible closed subset $Z \subset Y$ and every irreducible component $Z'$ of $f^{-1}(Z)$ we have $\mathrm{codim}(Z', X) = \mathrm{codim}(Z, Y)$
- $Y$ is ``equicodimensional'', $f$ maps closed points to closed points, and every non-empty fiber of $f$ has dimension $\dim X - \dim Y$.
Here ``equicodimensional'' means that every closed point has the same codimension.)
The basic problem here is the naming rather than the object itself. The name TCFT as far as I understand indicates that it's a topological field theory, whose origin is in conformal field theory. Namely there is a construction (a "topological twist") starting from an $N=2$ supersymmetric conformal field theory in two dimensions, that produces a topological field theory (in fact two, the A- and B-twists). This is a special case of a general theory of topological twists of SUSY quantum field theories, which is by far the predominant source of topological quantum field theories as far as I know, including many of the most interesting ones coming from SUSY gauge theories in four dimensions (these are sometimes called "TFTs of Witten type" as opposed to the very rare Schwarz type, like Chern-Simons theory, which come with a manifestly topological formulation).
Now when we say "TFT" here it is at a more refined chain level than the classical Atiyah-Segal axiomatic definition --- a synonym for TCFT in the sense of say Costello's beautiful paper on the subject is differential graded TFT. This means roughly that the theory is topological on a derived level -- its outputs are topologically invariant up to coherent homotopies. (This is the kind of refined topological invariance one always gets out of twisting SUSY field theory.)
What makes this very confusing initially is it seems conformal structures on a Riemann surface are playing an essential role: TCFT is defined in terms of chains on moduli spaces of complex structures. However this is a red herring (unless you are interested in the CFT origin of the construction) -- the moduli of complex structures is just playing the role of a nice model for the classifying space $BDiff(\Sigma)$ of topological surfaces, and everything can be said purely topologically (as it is in Hopkins-Lurie's work on the Cobordism Hypothesis). So really we are just defining a TFT on the chain level, in families (ie universally over moduli of topological surfaces). (This is a perspective I learned from Segal and Teleman and Freed and Costello, see Lurie's manuscript on TFTs for the contemporary perspective.)
Best Answer
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?