GRAPHS AND PROJECTIVE PLAINES IN 3-MANIFOLDS

ABSTRACT. Froper homotopy equivalent compact P:-irreducible and sufficiently large 3-manifolds are homeomorp,ic. The result is not kF,cwn for irreducible 3-manifolds that contain 2-sided projective planes. ven if one assumes the Poincareconjecture. i this paper to such a 3-mnifold M is associated a graph G(M) that specifies Low a maximal system of mutually disjoint non-isotopic projective pla,es is embedded in M; ad it is sho:; that G(M) is an invaria:t of the homotupy type cf M. On the uth hand it is shown that ay giveF, graph can be realzed as G(M) for infir.itely man;, irreducible and boundary irreducible M. As an applicatior, it is shown that any closed irreducible 3-manifold M that contains 2-sidle projective planes c; be obtied from a P’-iFreducible 3-manifold nO p ’ x S by removing a so!d Klein bottle from ech and gluing egether resulting bcuFdaries " furthermoare, ccrtains an orientation preservir,g simple closed cure such that eny nontrivial Dehn surgery along yields a p2-irreducihl-mani fold