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

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

2005-04-12

arxiv.org/abs/math/0504251v1

Mathematics

Geometric Topology

8 pages, 11 figures

Scientific paper

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.

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