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Posets, regular CW complexes and Bruhat order. (English) Zbl 0538.06001
A poset P is thin if all its intervals of length two have cardinality four. For other terminology see e.g., the author and M. Wachs [Trans. Am. Math. Soc. 277, 323-341 (1983; Zbl 0514.05009)]. In this very interesting paper the author continues his efforts at the classification of those posets which are somehow (combinatorially) topological in nature. In this case the posets are the ones obtained from CW-complexes K. Indeed, if K is such a complex then F(K) is the set of closed cells ordered by containment and augmented by a bottom element \(\hat0\). A d-CW- complex is a finite regular CW-complex such that every closed cell is a face of a d-dimensional cell. Shellability is defined for d-CW-complexes.
The author demonstrates that if P is a thin, nontrivial, finite, pure poset of length \(d+1\) with least element \(\hat0\), then P shellable \(\Rightarrow\) \(P\cong F(K)\), K a d-CW-complex;
\^P\(=P\cup \{\hat 0\}\) dual CL-shellable \(\Leftrightarrow\) K can be a shellable d-CW-complex;
\^P shellable and thin \(\Rightarrow\) K can be a d-CW-complex homeomorphic to the d-sphere;
\^P dual shellable and thin \(\Leftrightarrow\) K can be a shellable d-CW- complex homeomorphic to the d-sphere.
Using these results and observations applied to Coxeter groups and Bruhat orderings, the author sharpens some of his own results. He demonstrates that every Coxeter group (W,S) determines a topological space with a regular cell decomposition such that the Bruhat ordering of W is isomorphic to the incidence ordering of the cells. The associated space is a sphere when W is finite and contractible when W is infinite.
Reviewer: J.Neggers

06A06 Partial orders, general
06B30 Topological lattices
57Q05 General topology of complexes
57N99 Topological manifolds
52Bxx Polytopes and polyhedra
Full Text: DOI
[1] Björner, A., Shellable and Cohen-Macaulay partially ordered sets, Trans. amer. math. soc., 260, 159-183, (1980) · Zbl 0441.06002
[2] A. Björner: Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Advances in Math. (to appear).
[3] Björner, A.; Garsia, A.M.; Stanley, R.P., An introduction to Cohen-Macaulay partially ordered sets, (), 583-615
[4] Björner, A.; Wachs, M., Bruhat order of Coxeter groups and shellability, Advances in math., 43, 87-100, (1982) · Zbl 0481.06002
[5] Björner, A.; Wachs, M., On lexicographically shellable posets, Trans. amer. math. soc., 277, 323-341, (1983) · Zbl 0514.05009
[6] Borel, A.; Tits, J., Compléments à particle ‘groupes réductifs’, LH.ES. publ. math., 41, 253-276, (1972) · Zbl 0254.14018
[7] Bourbaki, N., Groupes et algèbres de Lie, chap 4, 5 et 6, Éléments de mathlématique, fasc. XXXIV, (1968), Hermann Paris · Zbl 0483.22001
[8] R. Brown: private communication.
[9] Danaraj, G.; Klee, V., Shellings of spheres and polytopes, Duke math. J., 41, 443-451, (1974) · Zbl 0285.52003
[10] J. Edmonds, K. Fukuda, A. Mandel: Topology of oriented matroids, in preparation.
[11] Farmer, F.D., Cellular homology for posets, Math. japonica, 23, 607-613, (1979) · Zbl 0416.55003
[12] Farmer, F.D., Homotopy spheres in formal language, Studies in applied math., 66, 171-179, (1982) · Zbl 0511.55009
[13] A.M. Garsia, D. Stanton: Group actions on Stanley-Reisner rings and invariants of permutation groups, Advances in Math. (to appear). · Zbl 0561.06002
[14] Grünbaum, B., Convex polytopes, (1967), Wiley London · Zbl 0163.16603
[15] Lindström, B., Problem P73, Aequationes math., 6, 113, (1971)
[16] Lundell, A.T.; Weingram, S., The topology of C,W complexes, (1969), Van Nostrand New York · Zbl 0207.21704
[17] Maunder, C.R.F., Algebraic topology, (1970), Van Nostrand New York · Zbl 0205.27302
[18] Spanier, E.H., Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303
[19] Walker, J.W., Topology and combinatorics of ordered sets, Thesis, (1981), M.I.T. Cambridge, MA
[20] Walker, J.W., A poset which is shellable but not lexicographically shellable, (1982) · Zbl 0579.06001
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