Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Titus' homotopies of normal curves


Author: George K. Francis
Journal: Proc. Amer. Math. Soc. 30 (1971), 511-518
MSC: Primary 55.70; Secondary 30.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0285009-5
MathSciNet review: 0285009
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A regular homotopy of a plane immersion of the circle, each of whose stages is a normal curve with the exception of a finite number of stages of the homotopy presenting a nonnegative convex double point self tangency or a transverse triple point, preserves the number of topologically inequivalent extensions of the immersion to an orientation preserving immersion of the disk. Extensions to properly interior mappings of the disk are similarly investigated.


References [Enhancements On Off] (What's this?)

  • [1] George K. Francis, The folded ribbon theorem. A contribution to the study of immersed circles, Trans. Amer. Math. Soc. 141 (1969), 271-303. MR 39 #4863. MR 0243542 (39:4863)
  • [2] -, Regular homotopies in general position, Differential Geometry Seminar Notes, University of Illinois, Urbana, Illinois, 1970.
  • [3] -, Extensions to the disk of properly nested plane immersions of the circle, Michigan Math. J. 17 (1970), 377-383. MR 0284985 (44:2209)
  • [4] André Gramain, Bounding immersions of codimension 1 in Euclidean space, Bull. Amer. Math. Soc. 76 (1970), 361-364. MR 0259931 (41:4560)
  • [5] Morris L. Marx, Normal curves arising from light open mappings of the annulus, Trans. Amer. Math. Soc. 120 (1965), 46-56. MR 33 #3278. MR 0195073 (33:3278)
  • [6] -, The branch point structure of extensions of interior boundaries, Trans. Amer. Math. Soc. 131 (1968), 79-98. MR 36 #5914. MR 0222864 (36:5914)
  • [7] Marston 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)
  • [8] Valentin Poénaru, Extensions des immersions en codimension 1 (d'après Samuel Blank), Séminaire Bourbaki 1967/68, Exposé 342, Benjamin, New York, 1969. MR 41 #3.
  • [9] Charles J. Titus, The combinatorial topology of analytic functions on the boundary of a disk, Acta Math. 106 (1961), 45-64. MR 29 #3652. MR 0166375 (29:3652)
  • [10] Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276-284. MR 1556973

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55.70, 30.00

Retrieve articles in all journals with MSC: 55.70, 30.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0285009-5
Keywords: Branch points, general position, intersection sequence, lifting arcs, light-open maps, monotone homotopy, normal immersions, properly interior maps, regular homotopy, topological equivalence
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society