A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions 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 facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.