Publication: Discrepancy between the rank and the Heegaard genus of a 3-manifold. (Spanish: La discrepancia entre el rango y el genero de Heegaard de una 3-variedad)
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1989
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Università del Salento
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Abstract
Let M be a closed orientable 3-manifold. The rank of M, rk(M), is the minimum number of elements that can generate π1(M). Clearly rk(M)≤Hg(M), where Hg(M) is the Heegaard genus of M. F. Waldhausen conjectured that equality holds here, but then. Boileau and H. Zieschang [Invent. Math. 76 (1984), no. 3, 455–468;] found an infinite set of (Seifert) manifolds M with Hg(M)=3 and rk(M)=2. To prove the latter equality they started with the presentation of π1(M) resulting from a Heegaard splitting of M of genus 3; they then reduced this presentation to a presentation with only two generators. In the present paper the author shows that one can perform such a reduction by using only Nielsen moves (which are in general not sufficient for transforming an arbitrary finite presentation to any other finite presentation of the same group). Actually he proves a more general theorem about reducing, by Nielsen moves alone, the number of generators in certain presentations coming from Heegaard splittings, and because of this theorem he believes that the strict inequality rk(M)<Hg(M) is quite commonplace among closed orientable 3-manifolds.
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Conference on Differential Geometry and Topology (Italian) (Lecce, 1989).
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M. BOILEAU, H. ZIESCHANG, Heegaard genus of closed orientable Seifert manifolds, Invent.Math.76(1984) 455-468.
R. CRAGGS, The Poincare conjecture is equivalent to a restrictedform ofthe Andrews-Curtis conjecture,(preprint).
R. CRAGGS, Freely reducmg groups rwding for 2 -compleus in 4-manifolds, Topology 28 (1989) 247 -27l.
R. CRAGGS, On links in 3-manifolds as obslructwns infree reduction problems, (preprint).
W. HAKEN, Various aspects of lhe 3.dimennsional Poincare problem. Topology of Manifolds, Proc.Inst Univ. of Georgia, Athens. Ga. (1969) 140-152, Markham, Oticago 1970.
J.M. MONTESINOS, Nole on a resull of Boileau-Zieschang, Lectures Notes Series of the London Math. Soc.,1l2 (1986), 241-252.
J.M. MONTESINOS, Heegaarddiagrams for closed 4 -manifolds. Geometric Topology, J.C. Cantrell,ed. Proc. Inst. Univ. of Goorgia, Athens, Ga, (1977), 219-237, Academic Press, New York (1979).
J.M. MONTESINOS, Genero de Heegaord y rango de una 3-variedad, Libro homenaje a A. Plans Zaragoza (1989).
W. METZLER, uber den Homotopyetyp zweidimensionaler CW-Komplexe una Elementransformationnen bei Darslellungen von Gruppen durch Erzengende und definierende Relationen, J. Reine angew. Math. 235 (1976)7-23.
J. SINGER, Three dimensional manifolds and their Heegaard diagrams, Trans. AMS 35 (1933) 88-111.
F. WALDHAUSEN, Some problems on 3-manifolds, Proc. of Symposia in Pure Math. 32 (1978) 313-322.