  We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz
on the rational algebraic $K$-theory of spaces coincide. This gives a way to
compute the positive and negative eigenspaces of the involution on rational
homotopy groups of pseudoisotopy spaces from the involution on rational
$S^{1}$-equivariant homology group of the free loop space of a simply-connected
manifold. As an application, we give explicit dimensions of the open manifolds
$V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of
complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy
groups.
