Local Homeomorphism Research Articles

Differentiable monotone maps on manifolds

Let Mn and Nn be closed manifolds, and let G be any (nonzero) module. (1) If f: M3 -N3 is C3 G-acyclic, then there is a closed C3 3-manifold K3 such that N3 # K3 is diffeomorphic to M3, and f1(y) is cellular for all but at most r points y E N3, where r is the number of nontrivial G-cohomology 3-spheres in the prime decomposition of K3. (2) Iff: M3 _> M3 orf: S3 _* M3 is G-acyclic, then f is cellular. In case G is Z or Z, (p prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If f: Mn -* Mn or f: Sn * Mn is real analytic monotone onto, then f is a homeomorphism. 1. Conventions and definitions. Let G be a module, and let A c Mn be compact. Then A is acyclic (k-acyclic; cellular) is the reduced Cech cohomology module H*(A; G)=0 (Hk(A;G)=O; there are n-cells AkcMMn such that Ak+lcint Ak and nk Ak=A). A proper onto map f: Mn -Nn is monotone (acyclic; cellular) if, for each y E Nn, f -1(y) is connected (acyclic; cellular). The branch set Bf is the set of points in Mn at which f fails to be a local homeomorphism. Standing hypotheses. Unless otherwise specified, all manifolds are connected, separable, and without boundary. Whenever the statement of a theorem refers to a map f without specifying its domain and range, it is understood that f: Mn -Nn is proper and onto. Whenever coefficients of cohomology are not specified, any (nonzero) module G may be used. Other conventions and definitions are as in [6, pp. 185-186]. For other work on monotone, acyclic, and cellular mapsf: Mn Nn, see bibliographies of this paper, [1], [15], and (vast) [30]. 2. Real analytic monotone maps. 2.1. DEFINITION. Let G be a module, and let Cf = Cf(G) = f I(Cl {y E Nn: fl*(f 1(y); G) # 0}). 2.2. LEMMA. Let L be a principal ideal domain, let Mn be orientable over L, and let dim (f(Cf)) < 0. Received by the editors August 26, 1970. AMS 1970 subject classifications. Primary 57A60, 54C10; Secondary 57D35, 32C05.

Local homeomorphisms of Euclidean space onto arbitrary manifolds.

Local homeomorphisms of Euclidean space onto arbitrary manifolds.

Differentiable open maps on manifolds

Introduction. This paper contains a detailed discussion with proofs of results announced in [4]. Let Mn and Nn be n-manifolds without boundary, and let f: Mn -+ N n be continuous. The map f is open if, whenever U is open in Mn, f(U) is open in Nn; it is light if, for every y e Nn ,dim (f'(y)) < 0. For n ? 2 there is a canonical light open map Fn,d: E giveni by Fn,d(Xl, x2, ..., Xn) = (u1, u2, x3 *, xv) where Ul + iU2 =(Xl t+ iX2 )d (i =Vi-1; d= 1,2, ... ). For n = 2 it is well known that a nonconstant complex analytic function is open and light. Conversely, Stoilow [12] proved that every light open map is locally topologically equivalent to an analytic map, and thus to some F2,d (d = 1,2, ... ). In fact (1.10), if M2 is compact and f is C2 and open, then f has this canonical structure. The main object of this paper is to prove (2.1) that the corresponding conclusion holds for arbitrary n (n _ 2), if we first remove an exceptional set of dimension at most n 3. Examples are given, especially in ?3, showing that the exceptional set and some of the hypotheses used are necessary. DEFINITION. As in [5] the branch set Bf is the set of points in Mn at which f fails to be a local homeomorphism. NOTATION. Iff : E nEP is C', then fi will be the ith component real-valued function, and Djfi will be the first partial derivative offi with respect to its jth coordinate. If y is a point in En, then yi will be its ith coordinate. The symbols Mn and NP will refer to manifolds of dimensions n and p, respectively. The statement that f: Mn -+ Np is Cm will imply that the manifolds are also Cm. The set of points in Mn at which the Jacobian matrix of f has rank at most q will be denoted by Rq. The closure of a set X is denoted by Cl[X] or X, its interior by int X, and the

Universal and special problems

The purpose of this paper is to investigate a situation which is best described by the following examples. I. Call a connected, locally connected and locally simply" connected topological space with base-point perfectly connected in brief. By a covering we mean a base-point preserving, surjective local homeomorphism of a perfectly connected topological space into a like one. Let ~0 be a ut,iversal covering of e'. Then 9 plays a double r61e. 1 ~ To each base-point preserving, continuous mapping ~0' of a simply and perfectly connected topological space into e', there exists one and only one base-point preserving, continuous function [ rendering the diagram

A lemma in the theory of structural stability of differential equations

The object of this note is to answer this question in the affirmative when F(x) is of class C2. (It can be mentioned that, even if F(x) is analytic, there need not exist such a map R of class C' with nonvanishing Jacobian; [2].) The mapping (1.3) to be obtained below maps solution paths of (1.1) into those of (1.4) preserving parametrizations. The first part of the paper concerns the linearization of a local homeomorphism of a Euclidean space into another of the same dimension. The second part concerns (1.1) or rather the group of homeomorphisms associated with (1.4). The reduction of the problem to mappings is suggested by Sternberg [3 ].

The Structure of Local Homeomorphisms, III

In this paper, we continue the study of local homeomorphisms in Euclidean n-space begun in [13] and [14]. The main result to be proved here is that any C- volume preserving transformation defined in some neighborhood of the origin, keeping the origin fixed, whose Jacobian matrix at the origin satisfies a formal condition (**) to be given below, can be brought by a volume preserving change of coordinates to a certain normal form. The set of germs of these normal forms fall into a finite number of classes, each of which constitutes a maximal commutative subgroup of the group of local Cvolume preserving maps. Furthermore, the condition (**) will be interpreted as a regularity condition. Thus, we will establish for the group of local C- volume preserving maps a theorem analogous to the one which asserts

On Ramified Riemann Domains

Let ϕ be a holomorphic mapping of an n-dimensional analytic space E into Cn. If ϕ is non-degenerate at every point of E, we call the pair (E, ϕ) a Riemann domain. The notion of a Riemann domain is a generalization of the notion of a concrete Riemann surface. A Riemann domain (E, ϕ) is said to be unramified if ϕ is a local homeomorphism, and to be ramified if otherwise.

On the Structure of Local Homeomorphisms of Euclidean n-Space, II

On the Structure of Local Homeomorphisms of Euclidean n-Space, II

Covering spaces, fibre spaces, and local homeomorphisms

Covering spaces, fibre spaces, and local homeomorphisms