Shellability and Higher Cohen-macaulay Connectivity of Generalized Cluster Complexes

Let Φ be a finite root system of rank n and let m be a positive integer. It is proved that the generalized cluster complex ∆m(Φ), introduced by S. Fomin and N. Reading, is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. This statement was conjectured by V. Reiner. More precisely, it is proved that the simplicial complex obtained from ∆m(Φ) by removing any subset of its vertex set of cardinality not exceeding m is pure, of the same dimension as ∆m(Φ), and shellable. An analogous statement is shown to hold for the positive part ∆m + (Φ) of ∆m(Φ). Finally, an explicit homotopy equivalence is given between ∆m + (Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.