# Definition of a contractible simplicial complex without appealing to topological realization

Let $\Delta$ be a simplicial complex. I believe the standard definition of $\Delta$ being contarctible is if the topological realization $|\Delta|$ is contractible. However, I am seeking a definition of a contractible simplicial complex that is in terms of the simplicial complex qua simplicial complex, possibly having something to do with its simplicial homology. In other words, something in terms of the faces or its simplicial homology or something that can be checked while staying in the category of simplicial complexes. Call such a definition $S$-contractible. I would then hope that a simplicial complex is $S$-contractible if and only if $|\Delta|$ is contractible. Is anyone aware of such a definition and if so, has it been shown to agree with the standard definition of contractible? Many thanks in advance.

-
If you use the more flexible language of simplicial sets, then there is a purely combinatorial notion of homotopy which can be used to express contractibility (and then a (Kan) simplicial set is contractible if and only if its geometric realization is). –  Akhil Mathew Jun 10 '12 at 23:06

## 1 Answer

You want the simplicial approximation theorem. One part, or perhaps I should say version, says that any continuous map between the geometric realizations of two simplicial complexes is homotopic to a simplicial map (possibly after subdivision). Another part/version says that there is a simplicial definition of what it means for two simplicial maps to be homotopic and that it agrees with the usual definition (possibly after subdivision).

Given that, you can write down a simplicial definition of what it means for two simplicial complexes to be homotopy equivalent, and you can write down a simplicial definition of what it means for there to exist a map from a point to your simplicial complex which is a homotopy equivalence. That's contractibility by simplicial approximation. But maybe you want something that doesn't require subdivision?

-
If your question is more like "is there an algorithm which takes as input the combinatorial data specifying a simplicial complex and decides whether it is contractible or not," I have no idea. That seems like it could be hard. –  Qiaochu Yuan Jun 9 '12 at 0:19
@Sora According to cstheory.stackexchange.com/a/6530 , it is undecidable whether a given simplicial complex is contractible. –  David Speyer Jun 9 '12 at 0:36
Thanks for the replies. I'll look carefully at the simplicial approximation theorem to see if I can make it work. I guess what I had in mind comes from the analogy with a 1-dimesional complex. If we view a 1-dimensional simplicial complex as a graph $G$, then $|G|$ is contractible if and only if $G$ is a tree. There are lots of good and easy to understand definitions of a tree. Something like a higher dimensional analogue of a tree is what I was hoping for. –  Sora Jun 9 '12 at 1:51
@Sora: the 1-dimensional case is pretty misleading. 2-dimensional complexes include the presentation complex (en.wikipedia.org/wiki/Presentation_complex) of any finitely presented group, and if such a complex is contractible then the group is necessarily trivial (although I do not think the converse is true). This is pretty close to a proof that contractibility is already not decidable for 2-complexes (since the question of whether a finitely presented group is trivial is well-known to be undecidable). –  Qiaochu Yuan Jun 9 '12 at 2:15