Date of Award

1980

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Abstract

In Kirby's problem list {R. Kirby: Problems in Low Dimensional Manifold Theory. Proc. Symp. Pure Math. 32, (1978), p.28} is "Problem 2.4: (Birman) Let (alpha) be the obvious homomorphism (eta)(,g)(' )(--->)(' )Aut ((pi)(,l)(N(,g))) where (eta)(,g) is the group of isotopy classes of orientation perserving homeomorphisms of N(,g). Is kernel ((alpha)) finitely generated?" Here N(,g) denotes the 3-dimensional orientable handlebody of genus g. See {J. Birman: Braids, Links and Mapping Class Groups. Ann. of Math. Studies No. 82, PUP(1975), p.220}. In {E. Luft: Actions of the Homeotopy Group of an Orientable 3-Dimensional Handlebody. Math. Ann. 234 (1978), Corollary 2.3} Luft proves that kernel ((alpha)) is generated by Dehn Twists along properly embedded 2-cells in N(,g). In {J. Birman: Private communication. Aug. 6, 1979} it was suggested that a geometric proof of Luft's result be found since Luft's proof was algebraic in nature. The author gives a constructive geometric proof of Luft's Theorem in the case of a handlebody of genus two.

Pages

65

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