Extensions of normal immersions of into

Author:
Morris L. Marx

Journal:
Trans. Amer. Math. Soc. **187** (1974), 309-326

MSC:
Primary 57D40; Secondary 30A90, 54C10

DOI:
https://doi.org/10.1090/S0002-9947-1974-0341505-0

MathSciNet review:
0341505

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that is an *immersion*, i.e., a map such that is never zero. We call *f normal* if there are only finitely many self-intersections and these are transverse double points. A normal immersion *f* can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of *f* to . In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows:

(1) When does *f* have a light open extension ? (Recall that *light* means is totally disconnected for all *y* and *open* means *F* maps open sets of the interior of to open sets of .) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism , such that *fh* has an analytic extension to ?

(2) Suppose that is light, open, sense preserving, and, at each point of , *F* is a local homeomorphism. At each point of the interior of , *F* is locally topologically equivalent to the power mapping on . Points where are called *branch points* and is the *multiplicity* of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of *f*. Also we can ask what multiplicities can arise for extensions of a given *f*.

(3) Given a normal *f*, find the maximum number of properly interior extensions of *f* that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank's solution of the immersion problem that prompted this paper.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0341505-0

Keywords:
Light open mapping,
normal immersion,
branch point multiplicity,
immersion of the two-disk

Article copyright:
© Copyright 1974
American Mathematical Society