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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

An equality for $ 2$-sided surfaces with a finite number of wild points


Authors: Michael D. Taylor and Harvey Rosen
Journal: Trans. Amer. Math. Soc. 167 (1972), 347-358
MSC: Primary 57A10
MathSciNet review: 0295315
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Abstract: Let S be a 2-sided surface in a 3-manifold that is wild from one side U at just m points. It is shown that the minimal genus possible for all members of a sequence of surfaces in U converging to S (where these surfaces each separate the same point from S in $ U \cup S$) is equal to the sum of the genus of S and a certain multiple of the sum of m special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of S and the value of m. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0295315-1
PII: S 0002-9947(1972)0295315-1
Keywords: 2-sided surfaces in 3-manifolds, surfaces with finitely many wild points, convergent sequence of surfaces, limiting genus, local enveloping genus, number of nonpiercing points of surfaces
Article copyright: © Copyright 1972 American Mathematical Society