====== Thomas Zaslavsky ======

====== Homotopy in Biased Graphs: Combinatorics vs. Topology ======

===== Abstract for the Combinatorics and Geometry/Topology Seminars 2004 January 29 =====

A path in the 1-skeleton of a topological cell complex (with endpoints in the 0-skeleton) is a sequence P = v<sub>0</sub>, e<sub>1</sub>, v<sub>1</sub>, e<sub>2</sub>, ..., v<sub>k</sub>. 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?