# Homotopies of edge paths in simplicial complexes

Suppose $K$ is a simplicial complex with base vertex $v_0$ and $p$ is an edge path in $K$ based at $v_0$ (so a sequence of vertices connected by edges starting and ending at $v_0$). In other words, $p$ is a simplicial map $p:M\to K$ where $M$ is some subdivision of $\partial\Delta_{1}$. If the loop $|p|:|\partial\Delta_{2}|\to |K|$ on realizations is null-homotopic is seems, according to this page, that there is a sequence of edge paths (where each step moves edge(s) across a 2-simplex) from $p$ to the constant edge path.

I would like to know if $|p|$ being null-homotopic also means there is a subdivision $N$ of the standard 2-simplex $\Delta_2$ so the subcomplex of $N$ corresponding to the boundary $\partial\Delta_{2}$ is precisely $M$ and a simplicial map $h:N\to K$ and $h$ restricted to $M$ is $p$? Perhaps this subdivision can be "built" out of the moves?

Of course, you can take a simplicial approximation to a homotopy $|\Delta_2|\to|K|$ that extends $|p|$, but what I am asking seems more combinatorial. If anyone can suggest texts/papers for me to look into that treat this question that would also be very nice!

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I'm a little more comfortable with simplicial sets, so let me try to answer your question for them.

Essentially, what you are asking is the following: let $K$ be a simplicial set, and let $p: \partial \Delta[2] \to K$ be a "loop" (simplicially) such that the geometric realization of $p$ is trivial. Then, after subdividing $K$ appropriately to some $K'$, you are asking whether one can "fill in" $p$ to get a map $\Delta[2] \to K$. The answer is yes, at least with Kan's notion of subdivision (see this article). The point is that, for a fibrant simplicial set (i.e. a Kan complex), a map $p: \partial \Delta[2] \to K$ is trivial on the geometric realizations precisely if it can be filled in to a map $\Delta[2] \to K$; that's essentially because of the (rather deep) equivalence between the homotopy categories of Kan complexes and CW complexes, which implies that a "simplicial" definition of the homotopy groups is the same as the homotopy groups of the realization.

In this case, the point becomes that subdividing infinitely gives a "fibrant replacement" for any simplicial set $K$, so one way to compute the homotopy groups of a realization of a simplicial set is to take the (combinatorial) homotopy groups of the infinite subdivision of $K$.

Incidentally, the whole story of the equivalence between the homotopy categories that I just mentioned is a very beautiful (and amazing) story in algebraic topology, due to Quillen. If you are interested in this sort of thing, there are a lot of sources that may be useful: for instance, Hovey's book on model categories, or Goerss-Jardine's "Simplicial homotopy theory." What Quillen essentially did was to formulate a notion of a "model category" (for homotopy theory) with essentially an axiomatization of obstruction theory; a model category has in particular a homotopy category (given by a suitable localization) which can be easily described. Classical methods essentially give you a model structure on topological spaces, but it is much harder and more subtle that there exists a model structure on simplicial sets encoding their homotopy theory. (Or rather, it's not too hard to construct a model structure, but it is certainly hard to show that it's what you want!) It's also subtle that these two model structures describe the same homotopy theory in a certain technical sense: they are Quillen equivalent.

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