Non-singular graph-manifolds of dimension 4

Abstract

A compact $4$–dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing $T^2$–bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy equivalence of closed oriented $4$–manifolds with reduced nonsingular graph-structures is homotopic to a diffeomorphism preserving the structures.