Pi-1 of Symplectyc Autimorphism Groups and Invertibles in by Seidel P.

By Seidel P.

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Vol. 4 remains true without any assumption on N ; this is one of the motivations for extending the theory beyond the case where (W + ) holds. As a final example, consider the case where M is four-dimensional. (W + ) holds for all symplectic four-manifolds. For N = 1, I¯ is vacuous. Assume that N ≥ 2 (this implies that (M, ω) is minimal). 1) holds unless (M, ω) is rational or ruled. For convenience, we reproduce the proof from [MS3]: for generic J, there are no non-constant J-holomorphic spheres w with c1 (w) ≤ 0 because the moduli space of such curves has negative dimension.

Birkh¨auser, Progress in Mathematics 133 (1995), 483–524. 1094 [M1] P. SEIDEL GAFA D. McDuff, Examples of symplectic structures, Invent. Math. 89 (1987), 13–36. [M2] D. McDuff, The local behaviour of holomorphic curves in almost complex 4-manifolds, J. Differential Geom. 34 (1991), 143–164. [M3] D. McDuff, Immersed spheres in symplectic 4-manifolds, Ann. Inst. Fourier 42 (1992), 369–392. [MS1] D. McDuff, D. Salamon, J-holomorphic Curves and Quantum Cohomology, Amer. Math. , University Lecture Notes Series 6, (1994).

3) du(z) + Jz− ◦ du(z) ◦ j − = 0 ∂u + Jz− (u) ∂s ∂u − XH − (s, t, u) ∂t =0 for z ∈ D− , for z = (s, t) ∈ R− × S 1 . Let h be a Riemannian metric on M and f ∈ C ∞ (M, R) a Morse function. If c is a critical point of aH −∞ and y a critical point of f , we denote by 1082 P. 3) which converge to c and with u(z0 ) ∈ W s (y; f, h). In the generic case, M− (c, y; H − , J− ) is a manifold of dimension µH −∞ (c) − if (y), and the zero-dimensional spaces are again finite. For fixed (H ∞ , J∞ ) and (H −∞ , J−∞ ), we will denote the space of all (H + , J+ , H − , J− , f, h) by C(H ∞ , J∞ , H −∞ , J−∞ ).

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