Connectivity of \(h\)-complexes. (Q1427012)

If a simplicial complex \(\Delta\) has a shelling in which the unique minimal faces from the shelling steps form a subcomplex of \(\Delta\), then this subcomplex is called the \(h\)-complex of \(\Delta\) with respect to the considered shelling (referred to as an \(H\)-shelling). The author investigates the \(h\)-complex \(\Delta_n\) which results from the standard shelling for the order complex of a truncated Boolean algebra \(B_n-{\widehat 0, \widehat1}\). \textit{P. H. Edelman} and \textit{V. Reiner} conjectured in [Adv. Math. 106, No. 1, 36--64 (1994; Zbl 0857.57024)] that the homology in dimension \(i\) is nonzero, \(\widetilde H_i(\Delta_n,Z)\not= 0\) if and only if \((3i+5)/2\leq n \leq 3i+4\). The main result of the reviewed article is a proof of this conjecture. Some possible generalizations are also suggested.