  We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of
smooth compact simply connected oriented 4-manifolds M bounded by a collection
of 3-spheres. We show that if M contains CP^2 (with either orientation) as a
connected summand then \Gamma(M) is independent of the number of boundary
components. By repackaging classical results of Wall, Kreck and Quinn, we show
that the natural homomorphism from the mapping class group to the group of
automorphisms of the intersection form becomes an isomorphism after
stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next
consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as
above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We
identify the homotopy type of the classifying space of this category as the
Hermitian algebraic K-theory of the integers. We also comment on versions of
these results for simply connected spin 4-manifolds. Finally, we observe that a
related 4-manifold operad detects infinite loop spaces.
