On the Diffeomorphism Group of S 1 × S 2

The main result of this paper is that the group Diff(S×S) of diffeomorphisms S1×S2→S1×S2 has the homotopy type one would expect, namely the homotopy type of its subgroup of diffeomorphisms that take each sphere {x}×S to a sphere {y}×S by an element of the isometry group O(3) of S , where the function x,y is an isometry of S , an element of O(2) . It is not hard to see that this subgroup is homeomorphic to the product O(2)×O(3)×ΩSO(3) , this last factor being the space of smooth loops in SO(3) based at the identity. This has the same homotopy type as the space of continuous loops. These loopspaces have H2i(ΩSO(3);Z) nonzero for all i , so we conclude that Diff(S×S) is not homotopy equivalent to a Lie group. Diffeomorphism groups of surfaces and many irreducible 3-manifolds are known to be homotopy equivalent to Lie groups (often discrete groups in fact), and S×S is the simplest manifold for which this is not true. Via the Smale conjecture, proved in [H], the calculation of the homotopy type of Diff(S×S) reduces easily to a problem about making families of 2 spheres in S×S disjoint. To state the problem in slightly more generality, let M be a connected 3 manifold containing a sphere S ⊂ M that does not bound a ball in M , and let S×[−1,1] ⊂ M be a bicollar neighborhood of this S . Let E be the space of smooth embedding f :S2→M3 whose image does not bound a ball, and let E′ be the subspace of embeddings f for which there exists x ∈ [−1,1] with f(S) disjoint from {x}×S . The result we need is: