International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 551-560
doi:10.1155/S0161171286000698

Graphs and projective plaines in 3-manifolds

Abstract

Proper homotopy equivalent compact P2-irreducible and sufficiently large 3-manifolds are homemorphic. The result is not known for irreducible 3-manifolds that contain 2-sided projective planes, even if one assumes the Poincaré conjecture. In this paper to such a 3-manifold M is associated a graph G(M) that specifies how a maximal system of mutually disjoint non-isotopic projective planes is embedded in M, and it is shown that G(M) is an invariant of the homotopy type of M. On the other hand it is shown that any given graph can be realized as G(M) for infinitely many irreducible and boundary irreducible M.As an application it is shown that any closed irreducible 3-manifold M that contains 2-sided projective planes can be obtained from a P2-irreducible 3-manifold and P2×S1 by removing a solid Klein bottle from each and gluing together the resulting boundaries: furthermore M contains an orientation preserving simple closed curve α such that any nontrivial Dehn surgery along α yields a P2-irreducible 3-manifold.