Kalai’s squeezed spheres are shellable. (English) Zbl 0959.52012
In [Discrete Comput. Geom. 3, 1-14 (1988; Zbl 0631.52009)] G. Kalai constructed an extremely large class of simplicial spheres as the boundaries of shellable (squeezed) balls. By a result of U. Pachner [Discrete Math. 81, No. 1, 37-47 (1990; Zbl 0698.52003)] such spheres are piecewise linear.
The author proves that Kalai’s spheres are, in fact, shellable. In the case of the dimension \(d\) of the sphere being even an explicit shelling order of the facets is given. The result for the odd-dimensional case follows from combining a reverse lexicographic shelling of a squeezed \(d\)-ball with the previosuly constructed shelling order for its \((d-1)\)-dimensional boundary.
Although examples of non-shellable (and even non-constructible) spheres are known [e.g.see M. Hachimori and G. M. Ziegler [Math. Z. 235, No. 1, 159-171 (2000)], the result of the paper under review raises the question how large the class of non-shellable (or non-constructible) spheres is.
The author proves that Kalai’s spheres are, in fact, shellable. In the case of the dimension \(d\) of the sphere being even an explicit shelling order of the facets is given. The result for the odd-dimensional case follows from combining a reverse lexicographic shelling of a squeezed \(d\)-ball with the previosuly constructed shelling order for its \((d-1)\)-dimensional boundary.
Although examples of non-shellable (and even non-constructible) spheres are known [e.g.see M. Hachimori and G. M. Ziegler [Math. Z. 235, No. 1, 159-171 (2000)], the result of the paper under review raises the question how large the class of non-shellable (or non-constructible) spheres is.
Reviewer: Michael Joswig (Berlin)
