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This paper studies topological properties of the lattices of non-crossingpartitions of types A and B and of the poset of injective words. Specifically,it is shown that after the removal of the bottom and top elements (if existent)these posets are doubly Cohen-Macaulay. This strengthens the well-known factsthat these posets are Cohen-Macaulay. Our results rely on a new poset fibertheorem which turns out to be a useful tool to prove double (homotopy)Cohen-Macaulayness of a poset. Applications to complexes of injective words arealso included.

A poset fiber theorem for doubly Cohen-Macaulay posets and its applications to non-crossing partitions and injective words