Goodwillie polytopes: a self-dual family arising from the topology of loop spaces

The S^1equivariant homology of the free loop space LX = Map(S^1, X) is a natural setting for important operations in string topology and is closely connected to cyclic homology. In work motivated by this connection, two Koszul-dual chain models for LX appear: a “smallest” model and a larger model in which certain geometric compatibilities become visible. The underlying idea is surprisingly concrete. Adams’ construction regards a singular simplex as a parametrized family of paths from its first vertex to its last. Arranging simplices cyclically into a “necklace” and concatenating the resulting paths produces families of free loops. Applying Adams’ construction directly leads to freehedra; first decomposing the simplices using the Alexander–Whitney diagonal leads instead to a family of larger parameter spaces: the Goodwillie polytopes.
These polytopes are interesting mathematical objects in their own right, and their combinatorics give geometric form to algebraic properties of the Hochschild-type constructions that motivate them. We will describe and visualize their structure and the striking self-duality of their face lattices.