ON THE TOPOLOGY OF DIFFEOMORPHISMS OF SYMPLECTIC 4-MANIFOLDS
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For a closed symplectic 4-manifold X, let Diff0(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of Diff0(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {n1, n2, . . . , nk} and any non-negative integer m, there exists a closed symplectic (or K¨ahler) 4-manifold X with [수식]such that the homologies Hi of the quotient space Diff0(X)/Symp(X)over the rational coefficients are non-trivial for all odd degrees i = 2n1 −1, . . . , 2nk − 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati´c.
Abstract
For a closed symplectic 4-manifold X, let Diff0(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of Diff0(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {n1, n2, . . . , nk} and any non-negative integer m, there exists a closed symplectic (or K¨ahler) 4-manifold X with [수식]such that the homologies Hi of the quotient space Diff0(X)/Symp(X)over the rational coefficients are non-trivial for all odd degrees i = 2n1 −1, . . . , 2nk − 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati´c.
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