Group presentations corresponding to spines of $3$-manifolds. II

Abstract:

Let $\phi = \langle {a_1}, \ldots ,{a_n}|{R_1}, \ldots ,{R_m}\rangle$ denote a group presentation. Let ${K_\phi }$ denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form ${K_\phi }$ for some $\phi$, but that not every ${K_\phi }$ is a spine of a compact 3-manifold. Neuwirth’s algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether ${K_\phi }$ 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 ${K_\phi }$, where $\phi$ has a particular form (e.g., $\langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle$), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every ${K_\phi }$ 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.