  There are two natural simplicial complexes associated to the noncrossing
partition lattice: the order complex of the full lattice and the order complex
of the lattice with its bounding elements removed. The latter is a complex that
we call the noncrossing partition link because it is the link of an edge in the
former. The first author and his coauthors conjectured that various collections
of simplices of the noncrossing partition link (determined by the undesired
parking spaces in the corresponding parking functions) form contractible
subcomplexes. In this article we prove their conjecture by combining the fact
that the star of a simplex in a flag complex is contractible with the second
author's theory of noncrossing hypertrees.
