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On systems of curves orthogonal to a 3-dimensional handlebody

On systems of curves orthogonal to a 3-dimensional handlebody On Systems of Curves 0rthogonal to a 3-Dimensional Handlebody By H. B. GRIFFITHS Introduction It is the purpose of this paper to prove an intuitively obvious topolo- gical theorem which is useful in the MORSE Theory of 3-manifolds. The theorem concerns a "normal form" for a system fin of m mutually disjoint JORDAN curves on the surface ~X of a standard 3-dimensional handle- body X or "Tn" (= solid torus of genus n), where m ~ n. To state it, we assume that if a curve J e 6r cuts a handle H of X, then J  H is a single arc, while the J's are homologically independent on X; we write J~ _[_ ~. Then the theorem asserts: -- i/ ~r _]_ ~ then there is an automorphism o/X on itsel/ which carries the J's onto m standard meridan curves on ~X. A precise statement is given in 4.2 below. When n ~ 3, the Theorem follows from simple homology considera- tions. For larger n, however, our proof is long and tedious, and the plan is as follows. After establishing some notation in w 1 for standard handle- bodies and curves, we first (w 2) consider a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On systems of curves orthogonal to a 3-dimensional handlebody

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References (7)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02992388
Publisher site
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Abstract

On Systems of Curves 0rthogonal to a 3-Dimensional Handlebody By H. B. GRIFFITHS Introduction It is the purpose of this paper to prove an intuitively obvious topolo- gical theorem which is useful in the MORSE Theory of 3-manifolds. The theorem concerns a "normal form" for a system fin of m mutually disjoint JORDAN curves on the surface ~X of a standard 3-dimensional handle- body X or "Tn" (= solid torus of genus n), where m ~ n. To state it, we assume that if a curve J e 6r cuts a handle H of X, then J  H is a single arc, while the J's are homologically independent on X; we write J~ _[_ ~. Then the theorem asserts: -- i/ ~r _]_ ~ then there is an automorphism o/X on itsel/ which carries the J's onto m standard meridan curves on ~X. A precise statement is given in 4.2 below. When n ~ 3, the Theorem follows from simple homology considera- tions. For larger n, however, our proof is long and tedious, and the plan is as follows. After establishing some notation in w 1 for standard handle- bodies and curves, we first (w 2) consider a

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Nov 17, 2008

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