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Graphs and projective planes in 3-manifolds. (English) Zbl 0622.57007

The authors associate to an irreducible 3-manifold M containing 2-sided projective planes a graph that specifies how a maximal system of mutually disjoint nonisotopic projective planes is situated in M. They prove that this graph is an invariant of the homotopy type of the pair (M,\(\partial M)\). Also, any given graph can be realized by infinitely many irreducible and \(\partial\)-irreducible 3-manifolds. It is proved as an application that any closed irreducible 3-manifold M containing a 2-sided projective plane can be obtained from a \(P^ 2\)-irreducible 3-manifold and \(P^ 2\times S^ 1\) (here \(P^ 2\) is the projective plane) by removing a solid Klain bottle from each and gluing together the resulting boundaries.
Reviewer: N.V.Ivanov

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
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