Associahedron, cyclohedron and permutohedron as compactifications of configuration spaces - Lambrechts, Pascal; Turchin, Victor; Volic, Ismar (My University, 2010)
As in the case of the associahedron and cyclohedron, the permutohedron can also be defined as an appropriate compactification of a configuration space of points on an interval or on a circle. The construction of the compactification endows the permutohedron with a projection to the cyclohedron, and the cyclohedron with a projection to the associahedron. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disk, but are still contractible. We briefly explain an application of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss manifold calculus.
Keywords: Polytopes; Cyclohedron; Associahedron; Homotopy limit