Testing 3-manifolds for projective planes

Wolfgang H. Heil

Vol. 44 (1973), No. 1, 139–145

DOI: 10.2140/pjm.1973.44.139

Abstract

It is well known that a closed 3-manifold M contains a (piecewise linearly embedded) essential separating 2-sphere if and only if π₁(M) is a nontrivial free product. In this paper necessary and sufficient conditions, in terms of π₁(M), are given for the existence of a projective plane in M. If M is irreducible this condition is that π₁(M) be an extension of Z or a nontrivial free product by Z₂. In particular this provides a criterion for deciding which irreducible closed 3-manifolds are not P²-irreducible.

Mathematical Subject Classification

Primary: 57A10

Secondary: 55A05

Milestones

Received: 11 August 1971

Revised: 7 April 1972

Published: 1 January 1973