${\mathbb{Z}}_n$-manifolds in 4-dimensional graph-manifolds

Abstract

A standard fact about two incompressible surfaces in an irreducible 3-manifold is that one can move one of them by isotopy so that their intersection becomes ${\pi }_1$-injective. By extending it on the maps of some 3-dimensional ${\mathbb{Z}}_n$-manifolds into 4-manifolds, we prove that any homotopy equivalence of 4-dimensional graph-manifolds with reduced graph-structures is homotopic to a diffeomorphism preserving the structures. Keywords: graph-manifold, ${\pi }_1$-injective ${\mathbb{Z}}_n$-submanifold.

8 pages, 11 figures