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Collapsibility of $\Delta(\Pi _n)/\mathcal{S}_n$
and some related CW complexes

Author: Dmitry N. Kozlov
Journal: Proc. Amer. Math. Soc. 128 (2000), 2253-2259
MSC (2000): Primary 05E25
Published electronically: December 7, 1999
MathSciNet review: 1662257
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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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Additional Information

Dmitry N. Kozlov
Affiliation: Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden

Received by editor(s): August 6, 1998
Received by editor(s) in revised form: September 18, 1998
Published electronically: December 7, 1999
Communicated by: John R. Stembridge
Article copyright: © Copyright 2000 American Mathematical Society

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