A normal form in the homeotopy group of a surface of genus 2, with applications to 3-manifolds

Birman, Joan S.

doi:10.1090/S0002-9939-1972-0295308-X

A normal form in the homeotopy group of a surface of genus $2$, with applications to $3$-manifolds
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by Joan S. Birman

Proc. Amer. Math. Soc. 34 (1972), 379-384

DOI: https://doi.org/10.1090/S0002-9939-1972-0295308-X

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Abstract:

It is shown that elements in the homeotopy group of a closed, compact, orientable 2-manifold of genus 2 can be put into a unique normal form which allows them to be enumerated systematically. As an application, the class of 3-manifolds which admit Heegaard splittings of genus 2 are shown to be denumerable, and a procedure is given for enumerating presentations for their fundamental groups.

References

Joan S. Birman and Hugh M. Hilden, On the mapping class groups of closed surfaces as covering spaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 81–115. MR 0292082
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 14, Springer-Verlag, Berlin-Göttingen-New York, 1965. MR 0174618
F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235–254. MR 248801, DOI 10.1093/qmath/20.1.235
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
C. D. Papakyriakopoulos, A reduction of the Poincaré conjecture to group theoretic conjectures, Ann. of Math. (2) 77 (1963), 250–305. MR 145496, DOI 10.2307/1970216