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.