Mathematical Research Letters

Volume 6, Issue 6, November 1999 pp. 681-696.

$4$-manifolds with inequivalent symplectic forms and $3$-manifolds with inequivalent fibrations

Authors: Curtis T. McMullen and Clifford H. Taubes
Author institution: Harvard University

Summary: We exhibit a closed, simply connected 4-manifold $X$ carrying two symplectic structures whose first Chern classes in $H^2(X,\hbox{\fontsize{9}{8}$\mathbb Z$})$ lie in disjoint orbits of the diffeomorphism group of $X$. Consequently, the moduli space of symplectic forms on $X$ is disconnected. The example $X$ is in turn based on a 3-manifold $M$. The symplectic structures on $X$ come from a pair of fibrations $\pi_0, \pi_1 : M \arrow S^1$ whose Euler classes lie in disjoint orbits for the action of $\Diff(M)$ on $H_1(M,\hbox{\fontsize{9}{8}$\reals$})$. }

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