Showing papers on "Winding number published in 1975"
TL;DR: In this article, the authors define a combinatorial structure called a Tn-assemblage, which can be used to define a holomorphic map from a compact, oriented surface into an oriented surface without boundary.
Abstract: Let f: S -k E be a normal curve in the plane. The extensions of f to immersions of the disk with n handles (Tn) can be determined as follows. A word for f is constructed using the definitions of Blank and Marx and a combinatorial structure, called a Tn-assemblage, is defined for such words. There is an immersion extending f to Tn iff the tangent winding number of f is 1 2n and f has a Tn-assemblage. For each n, a canonical curve fn with a topologically unique extension to Tn is described (fo = Jordan curve). Any extendible curve with the minimum number (2n + 2 for n > 0) of self-intersections is equivalent to fn. Introduction. Let f: aAM N be a regular map (Cl -immersion) of the boundary of a compact, oriented surface into an oriented surface without boundary. Since M and N admit the structure of Riemann surfaces, the problem of the existence and topological classification of holomorphic maps F: M -N that extend f, FlaM = f, have been of interest to many mathematicians, notably M. Morse and M. Heins [141, [11]. According to Stollow-Whyburn theory [161, [221, a continuous map F that is light, open, and sense-preserving on M aM and a local homeomorphism relative to M on aM is topologically equivalent to a holomorphic mapping from M to N. Such maps are now said to be properly interior. If the map is branch point free (local homeomorphism) and C1 on the boundary, then it is equivalent to a Cl-immersion [J. Jewett, 9]; [J. H. C. Whitehead, 20]. Accordingly, this study will be presented entirely in the C1 context. Particular success has been obtained when the immersion f has its oriented image V] in the plane E or sphere S2 lying in completely general position (normal immersion); that is, [VI has a finite number of transverse self-intersections (nodes). Such curves comprise a dense-open subset of C1 (aM, N) in the Cl -topology [Whitney, 21], and were combinatorily classified by Titus [17] by means of their Whitney-Titus intersection sequence. In contrast to arbitrary Presented to the Society, April 28, 1973; received by the editors June 27, 1973 and, in revised form, March 15, 1974 AMS (MOS) subject classificatIons (1970). Primary 57D35. (1) This paper is part of the author's Ph.D. dissertation written at the University of Illlnois under Professor George K. Francis. Copyright ?) 1975. Anmerican Malltemtialical Soclety This content downloaded from 157.55.39.78 on Sun, 19 Jun 2016 05:55:08 UTC All use subject to http://about.jstor.org/terms
15 citations
TL;DR: A winding number algorithm for closed polygonal paths based on the notion of counting the number of oriented signed cuts of the negative x axis by the path is given in this article, which is justified by a theory of integer-valued analogues of the complex log function.
Abstract: A winding number algorithm for closed polygonal paths (not necessarily simple) based on the notion of counting the number of oriented \"signed cuts\" of the negative x axis by the path is given. The algorithm is justified by a theory of integer-valued analogues of the complex log function. The algorithm is much simpler than those of J. V. Petty [this journal, v. 27, 1973, pp. 333-337] and H. R. P. Ferguson [Notices Amer. Math. Soc, v. 20, 1973, p. A-211] and leads to faster computation.
3 citations
01 Feb 1975
TL;DR: In this paper, a characterization of pseudocones over S which are weakly chainable is given, and a simple geometric interpretation of the re-sult is that a pseudocone over S is weakly-chainable if and only if the ab-solute value of the winding number of any subarc of S around S is bounded by some m > 0.
Abstract: . A pseudocone over X is a compactification of (0, lj withremainder X. S is a circle. A characterization of those pseudocones overS which are weakly chainable is given. (A continuum is weakly chainableif and only if it is a continuous image of the pseudoarc.) Covering projec-tions and liftings are used, and a simple geometric interpretation of the re-sult is that a pseudocone over S is weakly chainable if and only if the ab-solute value of the winding number of any subarc of (0, lj around S is bounded by some m > 0.The following terminology will be used here. A continuum is a compactconnected metric space. / = [0, l]; A = (0, l]; 5 is the unit circle in the com-plex numbers. If X is a continuum, a pseudocone over X is a compactifica-tion of A with remainder X [l], [2]. R is the set of real numbers and Z isthe set of integers. A continuum X is acyclic it every continuous map /:
2 citations