Group presentations corresponding to spines of -manifolds. II

Authors:
R. P. Osborne and R. S. Stevens

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote a group presentation. Let denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form for some , but that not every is a spine of a compact 3-manifold. Neuwirth's algorithm (Proc. Cambridge Philos. Soc. **64** (1968), 603-613) decides whether 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 , 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 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.

**[1]**R. H. Bing,*Mapping a 3-sphere onto a homotopy 3-sphere*, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 89–99. MR**0219071****[2]**Joan S. Birman,*A normal form in the homeotopy group of a surface of genus 2, with applications to 3-manifolds*, Proc. Amer. Math. Soc.**34**(1972), 379–384. MR**0295308**, https://doi.org/10.1090/S0002-9939-1972-0295308-X**[3]**Joan S. Birman and Hugh M. Hilden,*The homeomorphism problem for 𝑆³*, Bull. Amer. Math. Soc.**79**(1973), 1006–1010. MR**0319180**, https://doi.org/10.1090/S0002-9904-1973-13303-8**[4]**Wolfgang Haken,*Various aspects of the three-dimensional Poincaré problem*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 140–152. MR**0273624****[5]**L. Neuwirth,*An algorithm for the construction of 3-manifolds from 2-complexes*, Proc. Cambridge Philos. Soc.**64**(1968), 603–613. MR**0226642****[6]**L. Neuwirth,*Some algebra for 3-manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 179–184. MR**0276978****[7]**R. P. Osborne and R. S. Stevens,*Group presentations corresponding to spines of 3-manifolds. I*, Amer. J. Math.**96**(1974), 454–471. MR**0356058**, https://doi.org/10.2307/2373554**[8]**Richard S. Stevens,*Classification of 3-manifolds with certain spines*, Trans. Amer. Math. Soc.**205**(1975), 151–166. MR**0358786**, https://doi.org/10.1090/S0002-9947-1975-0358786-0**[9]**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099**, https://doi.org/10.2307/1970594**[10]**E. C. Zeeman,*On the dunce hat*, Topology**2**(1964), 341–358. MR**0156351**, https://doi.org/10.1016/0040-9383(63)90014-4**[11]**Heiner Zieschang,*Über einfache Kurven auf Vollbrezeln*, Abh. Math. Sem. Univ. Hamburg**25**(1961/1962), 231–250 (German). MR**0149469**, https://doi.org/10.1007/BF02992929**[12]**H. Cišang,*Simple path systems on full pretzels*, Mat. Sb. (N.S.)**66 (108)**(1965), 230–239 (Russian). MR**0193633**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0488062-9

Keywords:
Compact orientable 3-manifold,
spine,
2-complex,
group presentation,
*P*-graph

Article copyright:
© Copyright 1977
American Mathematical Society