International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 551-560

Graphs and projective plaines in 3-manifolds

Wolfgang Heil1 and Seiya Negami2

1Department of Mathematics, Florida State University, Tallahassee 32306-3027, FL, USA
2Department of Information Science, Tokyo Institute of Technology, Oh-okayama, Meguro-Ku, Tokyo 152, Japan

Received 17 March 1986; Revised 18 April 1986

Copyright © 1986 Wolfgang Heil and Seiya Negami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.