A nice acyclic matching on the nerve of the partition lattice

Ralf Donau

Published 2012-04-12, updated 2015-08-28Version 3

The author has already proven that the space \Delta(\Pi_n)/G is homotopy equivalent to a wedge of spheres of dimension n-3 for all natural numbers n>=3 and all subgroups G<S_1 X S_{n-1}. We wish to construct an acyclic matching on \Delta(\Pi_n)/G that allows us to give a basis of its cohomology. This is also a more elementary approach to determining the number of spheres. Furthermore we give a description of the group action by an action on the spheres. We also obtain another result that we call Equivariant Patchwork Theorem.