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 \(H_1(M, \mathbb{R})\).