Abstract
Let X be the symplectic manifold S^2× S^2. In this paper, we discuss anti-symplectic involutions on X. We construct one equivalent class of anti-symplectic involutions on X from the equivalent class of anti-symplectic involutions on S^2.
Highlights
To study group actions on 4-dim manifolds, fixed point set is helpful
(Atiyah & Bott, 1968) study the relation between fixed point set of group actions and the induced group actions on homology of manifolds. (Atiyah, 1982) find for compact symplectic manifold and the n-torus hamiltonian actions on it, they can recover the cohomology of manifold by the cohomology of fixed point set
There are some studies about anti-symplectic involutions on symplectic manifolds
Summary
To study group actions on 4-dim manifolds, fixed point set is helpful. For example, (Atiyah & Bott, 1968) study the relation between fixed point set of group actions and the induced group actions on homology of manifolds. (Atiyah, 1982) find for compact symplectic manifold and the n-torus hamiltonian actions on it, they can recover the cohomology of manifold by the cohomology of fixed point set. By using fixed point set, (Karsho, 1995) study the classification of periodic hamiltonian flows on compact symplectic 4-manifold. Since lagrangian spheres could be the fixed point set of some involutions, the classification of anti-symplectic involutions on symplectic manifolds is naturally be a research focus. We try to study the classification of anti-symplectic involutions on rational manifolds S 2 × S 2. Let T be the circle action on S 2, σ1, σ2 be two anti-symplectic involutions on S 2 induced by two reflections compatible with T , the anti-symplectic involutions τσ , τσ2 on (S 2 × S 2, ω ⊕ ω) must be smooth equivalent, where τσ , τσ2 are defined as below τσi : S 2 × S 2 → S 2 × S 2.
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