Group presentations corresponding to spines of manifolds. II
Authors:
R. P. Osborne and R. S. Stevens
Journal:
Trans. Amer. Math. Soc. 234 (1977), 213243
MSC:
Primary 57A10; Secondary 55A05
MathSciNet review:
0488062
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Abstract: Let denote a group presentation. Let denote the corresponding 2complex. It is well known that every compact 3manifold has a spine of the form for some , but that not every is a spine of a compact 3manifold. Neuwirth's algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603613) decides whether can be a spine of a compact 3manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graphlike object, called a RRsystem (railroad system), is defined. To each RRsystem corresponds a whole family of compact orientable 3manifolds 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 3manifold can be obtained in this way. An equivalence relation on RRsystems is defined so that equivalent RRsystems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3manifolds arises from a particularly simple kind of RRsystem called a reduced RRsystem. Following Neuwirth, it is shown how to determine when a RRsystem gives rise to a collection of closed 3manifolds.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704880629
PII:
S 00029947(1977)04880629
Keywords:
Compact orientable 3manifold,
spine,
2complex,
group presentation,
Pgraph
Article copyright:
© Copyright 1977
American Mathematical Society
