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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Group presentations corresponding to spines of $3$-manifolds. II
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by R. P. Osborne and R. S. Stevens PDF
Trans. Amer. Math. Soc. 234 (1977), 213-243 Request permission

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.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 213-243
  • MSC: Primary 57A10; Secondary 55A05
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0488062-9
  • MathSciNet review: 0488062