Collapsibility of Δ(Π_{𝑛})/𝒮_{𝓃} and some related CW complexes

Abstract

Let Δ ( Π n ) \Delta (\Pi _n) denote the order complex of the partition lattice. The natural S n \mathcal {S}_n -action on the set [ n ] [n] induces an S n \mathcal {S}_n -action on Δ ( Π n ) \Delta (\Pi _n) . We show that the regular CW complex Δ ( Π n ) / S n \Delta (\Pi _n)/\mathcal {S}_n is collapsible. Even more, we show that Δ ( Π n ) / S n \Delta (\Pi _n)/\mathcal {S}_n is collapsible, where Π Δ \Pi _\Delta is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the S n \mathcal {S}_n -representation on H ∗ ( Δ ( Π n ) ) H_*(\Delta (\Pi _n)) .