  The main purpose of this paper is to prove that the Bruhat-Chevalley ordering
of the symmetric group when restricted to the fixed-point-free involutions
forms an $EL$-shellable poset whose order complex triangulates a ball. Another
purpose of this article is to prove that the Deodhar-Srinivasan poset is a
proper, graded subposet of the Bruhat-Chevalley poset on fixed-point-free
involutions.
