International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 3, Pages 551-560
Graphs and projective plaines in -manifolds
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 -irreducible and sufficiently large -manifolds are homemorphic. The result is not known for irreducible -manifolds that contain -sided projective planes, even if one assumes the Poincaré conjecture. In this paper to such a -manifold is associated a graph that specifies how a maximal system of mutually disjoint non-isotopic projective planes is embedded in , and it is shown that is an invariant of the homotopy type of . On the other hand it is shown that any given graph can be realized as for infinitely many irreducible and boundary irreducible .
As an application it is shown that any closed irreducible -manifold that contains -sided projective planes can be obtained from a -irreducible -manifold and by removing a solid Klein bottle from each and gluing together the resulting boundaries: furthermore contains an orientation preserving simple closed curve such that any nontrivial Dehn surgery along yields a -irreducible -manifold.