Collapsibility of

and some related CW complexes

Author:
Dmitry N. Kozlov

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2253-2259

MSC (2000):
Primary 05E25

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

Published electronically:
December 7, 1999

MathSciNet review:
1662257

Full-text PDF

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]**A. Björner,*Shellable and Cohen-Macaulay partially ordered sets*, Trans. Amer. Math. Soc.**260**(1980), 159-183. MR**81i:06001****[3]**A. Björner,*Subspace arrangements*, in ``First European Congress of Mathematics, Paris 1992'' (eds. A. Joseph et. al.), Progress in Math.**119**, Birkhäuser, 1994, pp. 321-370. MR**96h:52012****[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**18:61a****[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]**R. Forman,*Morse theory for cell complexes*, Adv. Math.**134**(1998), no. 1, 90-145. MR**99b:57070****[9]**S. Gelfand and Y. Manin,*Methods of homological algebra*, Translated from the 1998 Russian original, Springer, Berlin, 1996. MR**97j:18001****[10]**P. Hanlon,*A proof of a conjecture of Stanley concerning partitions of a set*, European J. Combin.**4**(1983), no. 2, 137-141. MR**85b:05020****[11]**D. Quillen,*Higher algebraic -theory*I, Lecture Notes in Mathematics, vol. 341, Springer-Verlag, 1973, pp. 85-148. MR**49:2895****[12]**G. Segal,*Classifying spaces and spectral sequences*, Inst. Hautes Études Sci. Publ. Math. No.**34**, 1968, pp. 105-112. MR**38:718****[13]**R. P. Stanley,*Some aspects of groups acting on finite posets*, J. Combin. Theory Ser. A**32**(1982), no. 2, pp. 132-161. MR**83d:06002****[14]**S. Sundaram,*The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice*, Adv. in Math.**104**(1994), 225-296. MR**96c:05189****[15]**J. Thévenaz, P. J. Webb,*Homotopy equivalence of posets with a group action*, J. Comb. Theory, Ser. A**56**(1991), no. 2, 173-181. MR**92k:20049****[16]**V. A. Vassiliev,*Complexes of connected graphs*, in: L. Corwin et al. (eds.),*The Gelfand Mathematical Seminar, 1990-1992*, Birkhäuser, Boston, pp. 223-235. MR**94h:55032**

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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

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