A path in the 1-skeleton of a topological cell complex (with endpoints in the 0-skeleton) is a sequence P = v0, e1, v1, e2, …, vk. An elementary homotopy of P consists of replacing a subpath P' by another path Q', with the same endpoints, so that P' union Q' is contractible. One can require that P' union Q' is a circle, i.e., homeomorphic to a 1-sphere. Call this topological homotopy.
If we replace the 1-skeleton by an arbitrary graph and the condition of contractibility by a list of allowed circles in the graph, we have combinatorial homotopy. This is the sort of homotopy involved in my recent characterization of associative multary quasigroups. Here the list of allowed circles has to satisfy a “linearity” condition; the combination of the graph and the linear class of allowed circles is called a “biased graph”. A particular lemma in the proof of the quasigroup theorem displays clearly the operation of combinatorial homotopy.
The questions are: what does a topologist know (or want to know) about combinatorial homotopy, and how similar and how different are topological and combinatorial homotopy?