Celebratio Mathematica — Gordon — Complete Bibliography

Complete Bibliography

Works connected to Wolfgang H. Heil

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C. M. Gordon and W. Heil: “Simply-connected branched coverings of \( S^3 \),” Proc. Am. Math. Soc. 35 : 1 (September 1972), pp. 287–288. MR 296930 Zbl 0257.55002 article

C. M. Gordon and W. Heil: “Cyclic normal subgroups of fundamental groups of 3-manifolds,” Topology 14 : 4 (1975), pp. 305–309. MR 400232 Zbl 0331.57001 article

Let \( M \) be a compact, connected, orientable 3-manifold. If \( M \) is a Seifert fibre space (other than \( S^3 \)), then \( \pi_1(M) \) has a non-trivial cyclic normal subgroup, namely that generated by the class of an ordinary fibre. If the orbit-surface (Zerlegungsfläche) of the fibring is orientable, then this subgroup is central. Conversely, Waldhausen has shown [1967] that if \( M \) is irreducible and sufficiently large (in the sense of [Waldhausen 1968]), and if \( \pi_1(M) \) has a non-trivial centre, then \( M \) has a Seifert fibring with orientable orbit-surface. By an argument based on Waldhausen’s we prove the following (which was announced in [Heil 1973]).

Let \( M \) be a compact, connected, orientable, irreducible and sufficiently large 3-manifold. If \( \pi_1(M) \) contains a (non-trivial) cyclic normal subgroup, then either

\( M \) is a Seifert fibre space or
there exists a closed surface \( F \) in \( M \) which separates \( M \) into two twisted line bundles over a non-orientable surface \( G \).