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.
@article{1274896208,
author = {Lambrechts, Pascal and Turchin, Victor and Voli\'c, Ismar},
title = {Associahedron, Cyclohedron and Permutohedron
as compactifications of configuration spaces},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {17},
number = {1},
year = {2010},
pages = { 303-332},
language = {en},
url = {http://dml.mathdoc.fr/item/1274896208}
}
Lambrechts, Pascal; Turchin, Victor; Volić, Ismar. Associahedron, Cyclohedron and Permutohedron
as compactifications of configuration spaces. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp. 303-332. http://gdmltest.u-ga.fr/item/1274896208/