Group presentations corresponding to spines of manifolds. III
Authors:
R. P. Osborne and R. S. Stevens
Journal:
Trans. Amer. Math. Soc. 234 (1977), 245251
MSC:
Primary 57A10; Secondary 55A05
MathSciNet review:
0488063
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Abstract: Continuing after the previous papers of this series, attention is devoted to RRsystems having two towns (i.e., to compact 3manifolds 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 be a group presentation corresponding to a spine of a compact orientable 3manifold, and let w be a relator of involving just two generators a and b. If w is cyclically reduced, then either (a) w can be ``written backwards" (i.e., if , then w is a cyclic conjugate of ), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If is a group presentation with two generators and if the corresponding 2complex is a spine of a closed orientable 3manifold then, is a spine of a closed orientable 3manifold if and (except for two minor cases) only if 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."
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704880630
PII:
S 00029947(1977)04880630
Keywords:
Compact orientable 3manifold,
spine,
2complex,
group presentation,
Pgraph
Article copyright:
© Copyright 1977
American Mathematical Society
