Spherical curves that bound immersed discs
Author:
George K. Francis
Journal:
Proc. Amer. Math. Soc. 41 (1973), 8793
MSC:
Primary 57D40
MathSciNet review:
0321112
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Abstract: Let be an immersion of the oriented circle in the oriented sphere. Let the image lie in general position and have tangent winding number with respect to some point in its complement. The extensions of to an orientation preserving immersion of the disc are classified up to topological equivalence by the assemblages induced by a star of rays from to the complementary components of the curve. Applications to the classification problem of stable maps between closed surfaces are also discussed.
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 K. Bailey, Extending closed plane curves to immersions of a disc with handles, Dissertation, University of Illinois, Urbana, Ill., 1973.
 [2]
 S. J. Blank, Extending immersions of the circle, Dissertation, Brandeis University, Waltham, Mass., 1967.
 [3]
 , Oral communication, 1972.
 [4]
 G. K. Francis, The folded ribbon theorem, A contribution to the study of immersed circles, Trans. Amer. Math. Soc. 141 (1969), 271303. MR 39 #4863. MR 0243542 (39:4863)
 [5]
 , Extensions to the disk of properly nested plane immersions of the circle, Michigan Math. J. 17 (1970), 377383. MR 44 #2209. MR 0284985 (44:2209)
 [6]
 , Titus' homotopies of normal curves, Proc. Amer. Math. Soc. 30 (1971), 511518. MR 44 #2233. MR 0285009 (44:2233)
 [7]
 , Branched and folded parametrizations of the sphere, Bull. Amer. Math. Soc. (to appear). MR 0350753 (50:3245)
 [8]
 M. L. Marx, Normal curves arising from light open mappings of the annulus, Trans. Amer. Math. Soc. 120 (1965), 4656. MR 33 #3278. MR 0195073 (33:3278)
 [9]
 , Light open mappings on a torus with a disk removed, Michigan Math. J. 15 (1968), 449456. MR 38 #2750. MR 0234433 (38:2750)
 [10]
 , Extensions of normal immersions of into , Trans. Amer. Math. Soc. (to appear).
 [11]
 M. L. Marx and R. Verhey, Interior and polynomial extensions of immersed circles, Proc. Amer. Math. Soc. 24 (1970), 4149. MR 40 #5879. MR 0252660 (40:5879)
 [12]
 M. Morse, Topological methods in the theory of functions of a complex variable, Ann. of Math. Studies, no. 15, Princeton Univ. Press, Princeton, N.J., 1947. MR 9, 20. MR 0021089 (9:20g)
 [13]
 V. Poenaru, Extensions des immersions en codimension 1 (d'apres Samuel Blank), Sem. Bourbaki 1967/68, Exposé 342.
 [14]
 C. J. Titus, A theory of normal curves and some applications, Pacific J. Math. 10 (1960), 10831096. MR 22 #5014. MR 0114189 (22:5014)
 [15]
 , The combinatorial topology of analytic functions on the boundary of a disc, Acta Math. 106 (1961), 4564. MR 29 #3652. MR 0166375 (29:3652)
 [16]
 H. Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276284. MR 1556973
 [17]
 , On singularities of mappings of Euclidean spaces, I. Mappings of the plane into the plane, Ann. of Math. (2) 62 (1955), 347410. MR 17, 518. MR 0073980 (17:518d)
 [18]
 G. Whyburn, Topological analysis, 2nd rev. ed., Princeton Math. Series, no. 23, Princeton Univ. Press, Princeton, N.J., 1964. MR 29 #2758. MR 0165476 (29:2758)
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DOI:
http://dx.doi.org/10.1090/S00029939197303211120
PII:
S 00029939(1973)03211120
Article copyright:
© Copyright 1973 American Mathematical Society
