Graphs and projective plaines in 3 -manifolds

Wolfgang Heil and Seiya Negami

International Journal of Mathematics and Mathematical Sciences, 1986, vol. 9, 1-10

Abstract:

Proper homotopy equivalent compact P 2 -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 P 2 -irreducible 3 -manifold and P 2 × S 1 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 P 2 -irreducible 3 -manifold.

Date: 1986
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/9/719731.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/9/719731.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:719731

DOI: 10.1155/S0161171286000698

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().