Surfaces minimizing area in their homology class and group actions on 3- manifolds. (English) Zbl 0715.57005

Hass, Joel

Math. Z. 199, No. 4, 501-509 (1988).

Minimal and least area surfaces have proved to be most useful in studying 3-manifolds. A closed orientable 3-manifold is called Haken if it is irreducible and contains a two-sided incompressible surface. In this paper, surfaces minimizing area in homology classes are employed to give interesting results when free finite group actions on Haken manifolds give Haken quotients. A key question in the classification of 3-manifolds is whether every irreducible 3-manifold with infinite fundamental group is finitely covered by a Haken manifold. Note that W. P. Thurston [“Geometry and topology of 3-manifolds”, Lecture Notes, Princeton Univ., Princeton, NJ, 1977/78] has given a heuristic argument that most of the former 3-manifolds are non-Haken. Suppose M is a Haken 3-manifold, G is a free finite group action on M and \(M_ 1=M/G\). The main result, Theorem 3-1, shows that if there is \(\alpha\neq 0\) in \(H_ 2(M,{\mathbb{Z}})\) such that \(g(\alpha)=\pm \alpha\) for each g in G, then \(M_ 1\) is Haken. Note that the assumption also needs to be included that \(\alpha\) is of infinite order, since the theorem is false for torsion elements. Typical applications are when \(H_ 2(M,{\mathbb{Z}})={\mathbb{Z}}\) or G is cyclic and \(H_ 2(M,{\mathbb{Z}})\) has odd rank. The proof is by a neat use of cut-and- paste, plus the Meeks-Yau round-off trick, applied to a surface of least area in the homology class \(\alpha\).

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57S17	Finite transformation groups

Keywords:

invariant homology class; surfaces minimizing area in homology classes; free finite group actions on Haken manifolds; irreducible 3-manifold with infinite fundamental group

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