Collapsibility of

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the order complex of the partition lattice. The natural -action on the set induces an -action on . We show that the regular CW complex is collapsible. Even more, we show that is collapsible, where 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 -representation on .

**[1]**E. Babson, A. Björner, S. Linusson, J. Shareshian, V. Welker,*Complexes of not -connected graphs*, Topology**38**(1999), 271-299. CMP**99:05****[2]**Anders Björner,*Shellable and Cohen-Macaulay partially ordered sets*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 159–183. MR**570784**, 10.1090/S0002-9947-1980-0570784-2**[3]**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****[4]**E. Babson, D. N. Kozlov,*Group actions on posets*, preprint, 1998.**[5]**G. E. Bredon,*Introduction to compact transformation groups*, Academic Press, 1972.**[6]**P. E. Conner,*Concerning the action of a finite group*, Proc. Nat. Acad. Sci. U.S.A.**42**(1956), 349–351. MR**0079272****[7]**E. M. Feichtner and D. N. Kozlov,*On subspace arrangements of type*, preprint 489/1995, TU Berlin, 21 pages, (to appear in the special FPSAC'96 issue of Discr. Math.).**[8]**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**, 10.1090/S0002-9939-99-04544-X**[9]**Sergei I. Gelfand and Yuri I. Manin,*Methods of homological algebra*, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR**1438306****[10]**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**, 10.1016/S0195-6698(83)80043-2**[11]**Daniel Quillen,*Higher algebraic 𝐾-theory. I*, Algebraic 𝐾-theory, I: Higher 𝐾-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR**0338129****[12]**Graeme Segal,*Classifying spaces and spectral sequences*, Inst. Hautes Études Sci. Publ. Math.**34**(1968), 105–112. MR**0232393****[13]**Richard P. Stanley,*Some aspects of groups acting on finite posets*, J. Combin. Theory Ser. A**32**(1982), no. 2, 132–161. MR**654618**, 10.1016/0097-3165(82)90017-6**[14]**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**, 10.1006/aima.1994.1030**[15]**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**, 10.1016/0097-3165(91)90030-K**[16]**V. A. Vassiliev,*Complexes of connected graphs*, The Gel′fand Mathematical Seminars, 1990–1992, Birkhäuser Boston, Boston, MA, 1993, pp. 223–235. MR**1247293**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
05E25

Retrieve articles in all journals with MSC (2000): 05E25

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

Email:
kozlov@math.ias.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05301-0

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