The stable mapping class group of simply connected 4-manifolds

Abstract

We consider mapping class groups Γ(M ) = π 0 Diff(M fix ∂M ) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3spheres. 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 2 #CP 2 . We next consider the 3+1 dimensional cobordism 2-category C of 3spheres, 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.1991 Mathematics Subject Classification. 57S05 (19G38, 57R90).

Stable isotopy in four dimensions

“…By Proposition 5.2, the resulting diffeomorphism of M # X is well-defined, possibly up to Dehn twists supported on an S 3 × I region separating the two summands. We claim (compare [35,Theorem 2.4]) that for X as indicated in the theorem, such a Dehn twist is isotopic to the identity. To see this, consider an S 1 action on X with a fixed point p; assume for convenience that it acts isometrically.…”

“…Similarly, two embedded 2-spheres S, T ⊂ X (respectively, diffeomorphisms f, g : X → X) are n-stably isotopic if the natural embeddings S, T ⊂ X # nS 2 ×S 2 are isotopic (respectively, the connected sums of f and g with the identity map 1 1 on nS 2 ×S 2 are isotopic; this connected sum is shown to be well-defined up to isotopy in Section 5; cf. [35]). There is also a weaker notion of n-stable equivalence, requiring only the existence of diffeomorphisms σ and τ of X # nS 2 ×S 2 such that T = σ(S) (respectively, g # 1 1 = τ • (f # 1 1) • σ).…”

The Dehn twist on a sum of two K3 surfaces

“…More Dehn twists. e non-triviality of δ in Proposition 1.2 is a question raised in a more general form by Giansiracusa in [8]. To describe this, let X be a simplyconnected, closed, spin 4-manifold, and let X (n) be obtained from X by removing n disjoint balls.…”

“…It is shown in [8] that the corresponding map on π 0 has kernel equal to either (Z/2) n−1 or (Z/2) n . e ambiguity results from the question of whether the particular diffeomorphism δ (n) , defined as the composite of the Dehn twists around the n disjoint spheres parallel to the boundary components, is isotopic to the identity in D (X (n) , ∂).…”

Spin Cobordism Categories in Low Dimensions

“…Similarly, Ω ∞ M T Spin(4) ∼ Z 2 × Ω ∞ M T Spin(4) with the classes of a K3 surface and the quaternionic projective plane as natural geometric generators for π 0 [11]. The Euler characteristic χ and the signature σ are a basis for the linear functionals on this group, at least over Z[1/2], and if χ * , σ * denote the dual basis elements, then…”