J. Korean Math. Soc. 2010; 47(4): 675-689
Printed July 1, 2010
https://doi.org/10.4134/JKMS.2010.47.4.675
Copyright © The Korean Mathematical Society.
Jin Hong Kim
Korea Advanced Institute of Science and Technology
For a closed symplectic 4-manifold $X$, let ${\rm Diff}_0(X)$ be the group of diffeomorphisms of $X$ smoothly isotopic to the identity, and let ${\rm Symp}(X)$ be the subgroup of ${\rm Diff}_0 (X)$ consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers $\{ n_1, n_2, \ldots, n_k \}$ and any non-negative integer $m$, there exists a closed symplectic (or K\"ahler) 4-manifold $X$ with $b_2^+(X)>m$ such that the homologies $H_i$ of the quotient space ${\rm Diff}_0 (X)/{\rm Symp}(X)$ over the rational coefficients are non-trivial for all odd degrees $i=2n_1-1, \ldots,2n_k-1$. 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.
Keywords: Seiberg-Witten invariants, symplectic diffeomorphism, symplectic structures
MSC numbers: Primary 57R57
2002; 39(1): 103-117
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