We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic
K
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
S^{1}
-equivariant homology groups of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds
V
V
that appear in Belegradek-Farrell-Kapovitch’s work for which the spaces of complete nonnegatively curved metrics on
V
V
have nontrivial rational homotopy groups.</p
