Associahedron, cyclohedron and permutohedron as compactifications of configuration spaces. (English) Zbl 1226.51004

Lambrechts, Pascal; Turchin, Victor; Volić, Ismar

Bull. Belg. Math. Soc. - Simon Stevin 17, No. 2, 303-332 (2010).

It is well-known that associahedra and cyclohedra can be defined as compactifications of configuration spaces. The authors show how 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. The authors show that the preimages of any point via these projections might not be homeomorphic to a disk, but are still contractible. They also briefly explain an application of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss manifold calculus.

Reviewer: David Chataur (Villeneuve d’Ascq)

MSC:

51M20	Polyhedra and polytopes; regular figures, division of spaces
57N25	Shapes (aspects of topological manifolds)
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

polytopes; cyclohedron; associahedron; homotopy limit; compactifications of configuration spaces; permutohedron; knot spaces; Goodwillie-Weiss manifold calculus