Symplectic Automorphisms of T ∗ S 2

Let T ⊂ T S be the bundle of unit discs, which is a compact symplectic manifold with boundary. Let S be the group of symplectic automorphisms of T which are trivial near ∂T . In [5] it was proved that π0(S) contains an element of infinite order. This element is given by the ‘generalized Dehn twist’ τ which is the monodromy map of a quadratic singularity [2] (see also [4]). The fact that [τ ] has infinite order is interesting because the class of τ in π0(D), where D is the corresponding group of diffeomorphisms, has order two. This note contains a short direct proof of the fact that S is weakly homotopy equivalent to the discrete group Z, with τ as a generator. The proof proceeds by compactifying T to S × S. The topology of the symplectic automorphism group of S × S was determined by Gromov [3], and we use a variant of his argument. Let T ⊂ T S be the space of cotangent vectors of length ≤ 1, and η its standard symplectic structure. Let D be the group of diffeomorphisms φ : T −→ T such that φ = id in a neighbourhood of ∂T , with the Ctopology, and S the subgroup of those φ which are symplectic (that is, they preserve η). Theorem 1. π0(S) ∼= Z, and πi(S) = 1 for all i > 0. The components of S cannot all be distinguished in D: Corollary 2. The image of the map π0(S) −→ π0(D) is isomorphic to Z/2. We will prove both results by compactifying (T, η) to S ×S with its standard symplectic structure ω (the one for which both spheres have equal areas). More precisely, one can identify (up to rescaling the symplectic forms) the interior of (T, η) with the complement of the diagonal ∆ in (S×S, ω). This has the following consequence: let D2 be the group of diffeomorphisms ψ of S ×S such that ψ|∆ = id and which act trivially on the normal bundle ν∆. Then D2 is w.h.e. (weakly homotopy equivalent) to D. Similarly, the subgroup S2 ⊂ D2 of those ψ which preserve ω is w.h.e. to S. Consider the larger group D1 ⊃ D2 of oriented diffeomorphisms of S 2 × S which are equal to the identity on ∆, and the subgroup S1 ⊂ D1 of those which are symplectic. Let G be the group of gauge transformations of ν∆ Date: 19/3/1998. Research supported by NSF grant DMS 9304580. 1