Group presentations corresponding to spines of -manifolds. II

Authors:
R. P. Osborne and R. S. Stevens

Journal:
Trans. Amer. Math. Soc. **234** (1977), 213-243

MSC:
Primary 57A10; Secondary 55A05

MathSciNet review:
0488062

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Abstract: Let denote a group presentation. Let denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form for some , but that not every is a spine of a compact 3-manifold. Neuwirth's algorithm (Proc. Cambridge Philos. Soc. **64** (1968), 603-613) decides whether can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length.

In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form , where has a particular form (e.g., ), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every which is a spine of some compact orientable 3-manifold can be obtained in this way.

An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1977-0488062-9

Keywords:
Compact orientable 3-manifold,
spine,
2-complex,
group presentation,
*P*-graph

Article copyright:
© Copyright 1977
American Mathematical Society