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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

##### MSC:
 06A06 Partial orders, general 06B30 Topological lattices 57Q05 General topology of complexes 57N99 Topological manifolds 52Bxx Polytopes and polyhedra
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