Categorified Associahedra and alternate blueprints for a free group element.
by
Stefan Forcey
Tennessee State University

The boundary of the nth associahedron K(n) is topologically equivalent to the (n-3)sphere. The boundary of the nth composihedron CK(n) is topologically equivalent to S^(n-2). One way of describing the indexing of vertices of the composihedra is by referring to equivalent binary lists of words, as opposed to binary lists of generators as in the associahedra. Another way is to refer to binary trees with weighted leaves, where the weights of the leaves sum to n. This generalizes the indexing of the associahedra by binary trees with n leaves. This last point of view allows us to count the vertices of the composihedra by the binomial transform of the Catalan numbers. It also allows us to construct a realization of CK(n) as a convex polytope, using methods recently developed by Loday for the associahedra. The new polytopes can be seen as a version of Stasheff's associahedra where sets have been replaced by objects in a general category.

When labeling vertices of the composihedra with bracketed lists of generators and words, we can view the nth polytope as representing an arbitrary reduced group element in the free group. Each vertex describes a sequence of concatenations for building that element from shorter ones. If the group is not free then the group element will instead be represented by a complex of polytopes, with faces identified in which the same rewriting of the same words has occurred between each vertex of those faces. The topology of such complexes is shown for some small examples.

Date received: March 17, 2006