4-Manifolds With Inequivalent Symplectic Forms and 3-Manifolds With Inequivalent Fibrations

Abstract

We exhibit a closed, simply connected 4-manifold (X) carrying two symplectic structures whose first Chern classes in (H^2 (X, \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 \rightarrow S^1) whose Euler classes lie in disjoint orbits for the action of ( \mathrm{Diff}(M) ) on (H_1(M, \mathbb{R})).