Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Group presentations corresponding to spines of $3$-manifolds. III
HTML articles powered by AMS MathViewer

by R. P. Osborne and R. S. Stevens PDF
Trans. Amer. Math. Soc. 234 (1977), 245-251 Request permission

Abstract:

Continuing after the previous papers of this series, attention is devoted to RR-systems having two towns (i.e., to compact 3-manifolds with spines corresponding to group presentations having two generators). An interesting kind of symmetry is noted and then used to derive some useful results. Specifically, the following theorems are proved: Theorem 1. Let $\phi$ be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of $\phi$ involving just two generators a and b. If w is cyclically reduced, then either (a) w can be “written backwards" (i.e., if $w = {a^{{m_1}}}{b^{{m_1}}}{a^{{m_2}}}{b^{{n_2}}} \ldots {a^{{m_k}}}{b^{{n_k}}}$, then w is a cyclic conjugate of ${b^{{n_k}}}{a^{{m_k}}} \ldots {b^{{n_2}}}{a^{{m_2}}}{b^{{n_1}}}{a^{{m_1}}}$), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If $\phi$ is a group presentation with two generators and if the corresponding 2-complex ${K_\phi }$ is a spine of a closed orientable 3-manifold then, ${K_\phi }$ is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if $\phi$ has two relators and among the six allowable types of syllables (3 in each generator), exactly four occur an odd number of times. Further, each of the two relators can be “written backwards."
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A10, 55A05
  • Retrieve articles in all journals with MSC: 57A10, 55A05
Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 245-251
  • MSC: Primary 57A10; Secondary 55A05
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0488063-0
  • MathSciNet review: 0488063