Collapsibility of $\Delta (\Pi _n)/\mathcal {S}_n$ and some related CW complexes
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- by Dmitry N. Kozlov PDF
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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 $H_*(\Delta (\Pi _n))$.References
- E. Babson, A. Björner, S. Linusson, J. Shareshian, V. Welker, Complexes of not $i$-connected graphs, Topology 38 (1999), 271–299.
- Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183. MR 570784, DOI 10.1090/S0002-9947-1980-0570784-2
- Anders Björner, Subspace arrangements, First European Congress of Mathematics, Vol. I (Paris, 1992) Progr. Math., vol. 119, Birkhäuser, Basel, 1994, pp. 321–370. MR 1341828
- E. Babson, D. N. Kozlov, Group actions on posets, preprint, 1998.
- G. E. Bredon, Introduction to compact transformation groups, Academic Press, 1972.
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- E. M. Feichtner and D. N. Kozlov, On subspace arrangements of type $\mathcal {D}$, preprint 489/1995, TU Berlin, 21 pages, (to appear in the special FPSAC’96 issue of Discr. Math.).
- Erkki Laitinen and Krzysztof Pawałowski, Smith equivalence of representations for finite perfect groups, Proc. Amer. Math. Soc. 127 (1999), no. 1, 297–307. MR 1468195, DOI 10.1090/S0002-9939-99-04544-X
- Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306, DOI 10.1007/978-3-662-03220-6
- Phil Hanlon, A proof of a conjecture of Stanley concerning partitions of a set, European J. Combin. 4 (1983), no. 2, 137–141. MR 705966, DOI 10.1016/S0195-6698(83)80043-2
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393, DOI 10.1007/BF02684591
- Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132–161. MR 654618, DOI 10.1016/0097-3165(82)90017-6
- Sheila Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math. 104 (1994), no. 2, 225–296. MR 1273390, DOI 10.1006/aima.1994.1030
- J. Thévenaz and P. J. Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991), no. 2, 173–181. MR 1092846, DOI 10.1016/0097-3165(91)90030-K
- V. A. Vassiliev, Complexes of connected graphs, The Gel′fand Mathematical Seminars, 1990–1992, Birkhäuser Boston, Boston, MA, 1993, pp. 223–235. MR 1247293
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, 100 44 Stockholm, Sweden
- Email: kozlov@math.ias.edu
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2253-2259
- MSC (2000): Primary 05E25
- DOI: https://doi.org/10.1090/S0002-9939-99-05301-0
- MathSciNet review: 1662257