• 1 Department of Mathematics [Athens]
• 2 Erwin Schrödinger Institute for Mathematical Physics
• 3 Fakultät für Mathematik [Wien]

In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homotopy Cohen-Macaulay posets, which can be considered as an extension of the classical poset fiber theorem for homotopy Cohen-Macaulay posets.