The Stable Mapping Class Group of Simply Connected 4-manifolds

We consider mapping class groups Γ(M) = π0Diff(M fix ∂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 or CP 2 as a connected summand then all Dehn twists around 3–spheres are trivial, and furthermore, Γ(M) is independent of the number of boundary components. By repackaging classical results in surgery and handlebody theory from 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 #CP . We next consider the 3+1 dimensional cobordism 2–category C 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.