Euler characteristic of the truncated order complex of generalized noncrossing partitions

The purpose of this paper is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized non crossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized non crossing partitions associated to well-generated complex reflection groups.