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

Kozlov, Dmitry

doi:10.1090/S0002-9939-99-05301-0

Collapsibility of $\Delta (\Pi _n)/\mathcal {S}_n$ and some related CW complexes
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by Dmitry N. Kozlov PDF

Proc. Amer. Math. Soc. 128 (2000), 2253-2259 Request permission

Abstract:

Let $\Delta (\Pi _n)$ denote the order complex of the partition lattice. The natural $\mathcal {S}_n$-action on the set $[n]$ induces an $\mathcal {S}_n$-action on $\Delta (\Pi _n)$. We show that the regular CW complex $\Delta (\Pi _n)/\mathcal {S}_n$ is collapsible. Even more, we show that $\Delta (\Pi _n)/\mathcal {S}_n$ is collapsible, where $\Pi _\Delta$ 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 $\mathcal {S}_n$-representation on $H_*(\Delta (\Pi _n))$.

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