Closed curves that never extend to proper maps of disks

Author:
J. Scott Carter

Journal:
Proc. Amer. Math. Soc. **113** (1991), 879-888

MSC:
Primary 57M35; Secondary 57N05, 57N10, 57Q35

DOI:
https://doi.org/10.1090/S0002-9939-1991-1070511-1

MathSciNet review:
1070511

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Abstract: If a closed curve in an orientable surface bounds a disk in a handlebody, then the double points on the boundary admit certain pairings that are called filamentations. Intersection numbers are associated to the filamentations; these numbers provide a necessary criterion for the existence of a disk bounded by a given curve. As an application, a closed curve with three generic double points in a surface of genus 2 is given that bounds no disk in a handlebody. This is the most simple example of a closed curve that does not bound a disk. The example is generalized to find for each , a closed curve with crossings that does not bound a disk.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1070511-1

Article copyright:
© Copyright 1991
American Mathematical Society