Vol. 44, No. 1, 1973

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Testing 3-manifolds for projective planes

Wolfgang H. Heil

Vol. 44 (1973), No. 1, 139–145

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
Received: 11 August 1971
Revised: 7 April 1972
Published: 1 January 1973
Wolfgang H. Heil