Quotients of the topology of the partition lattice which are not homotopy equivalent to wedges of spheres

TL;DR

This paper disproves a conjecture by showing that certain quotient topologies of the partition lattice are not homotopy equivalent to wedges of spheres, specifically for prime numbers p >= 5.

Contribution

It demonstrates that the quotient (ar{\u03a0}_p)/C_p is not homotopy equivalent to a wedge of spheres for any prime p  5, challenging previous assumptions.

Findings

(ar{}_p)/C_p is not homotopy equivalent to a wedge of spheres for prime p  5

Counterexample to the conjecture for all prime p  5

Disproves the general conjecture about (ar{}_n)/G homotopy types

Abstract

The reader of doi:10.1016/j.topol.2010.08.006 might conjecture that \Delta(\bar{\Pi}_n)/G is homotopy equivalent to a wedge of spheres for any n>=3 and any subgroup G<S_n. We disprove this by showing that \Delta(\bar{\Pi}_p)/C_p is not homotopy equivalent to a wedge of spheres for any prime number p>=5.