Covering properties
by
Cabiria Andreian Cazacu
University of Bucharest, Romania

Many results of complex function theory have been obtained by means of coverings. Without diminishing the role of the universal coverings, sometimes other coverings are also important, e.g. the double orienting covering of nonorientable Riemann (Klein) surfaces. By using the L.V. Ahlfors - L. Sario terminology, we deal in this paper with regular (unramified and unlimited) and in particular with normal (Galois) coverings ([X\tilde] , \Pi, X) and ([(X')\tilde] , \Pi', X') and with Hausdorff, linearly connected, locally (l.) simply connected and l. compact topological spaces, endowed with a metric by means of their universal coverings such that \Pi: [X\tilde] --> X and \Pi': [(X')\tilde] --> X' be l. isometries.

We establish lifting respectively factorizing properties of 1) continuous mappings and 2) homeomorphisms: f : X --> X' and [f\tilde] : [X\tilde] --> X' under a natural compatibility condition between the coverings.

E.g. in the case of liftings and regular coverings, we denote [p\tilde] in [X\tilde] , p in \Pi([p\tilde]), \Pi_[p\tilde] the homomorphism between the fundamental groups: \pi₁ ( [X\tilde], [p\tilde]) --> \pi₁ (X, p), G_{\Pi, [p\tilde]} = \Pi _[p\tilde][\pi₁( [X\tilde], [p\tilde])], p' = f(p), f_p:\pi₁ (X, p) --> \pi₁ (X', p').

The condition assuring the lifting [f\tilde] of f with [(p')\tilde] = [f\tilde] ( [p\tilde]) is for 1) f_p(G_{\Pi, [p\tilde]}) subset G_{\Pi', [(p')\tilde]} and for 2) f_p(G_{\Pi, [p\tilde]}) = G_{\Pi', [(p')\tilde]}. Then we prove:

Proposition 1. Let {f_n } be a sequence of continuous mappings f_n : X --> X' and [f\tilde]_n : [X\tilde] --> [(X')\tilde] a lifting of f_n. If the sequence {[f\tilde]_n } l. uniformly (u.) converges to [f\tilde]₀, then [f\tilde]₀ is the lifting of a continuous mapping f₀ and {f_n } l.u. converges to f₀.

Proposition 2. If {f_n } l.u. converges to f₀ and [f\tilde]_n , [f\tilde] ₀ are liftings of f_n, respectively f₀, such that there is a point [p\tilde]₀ in [X\tilde] for which {[f\tilde]_n ([p\tilde]₀) } tends to [f\tilde]₀ ([p\tilde]₀), then { [f\tilde]_n } l.u. converges to [f\tilde]₀.

Corollary. Let be: p, p', [p\tilde], [(p')\tilde] arbitrarily fixed points, F a family of mappings f and [(F)\tilde] a family of liftings [f\tilde] as before. Then F is normal (closed) iff [(F)\tilde] is normal (respectively, closed).

The paper contains results on case 2) of homeomorphisms and also on normal coverings.

This method was applied in joint papers with Victoria Stanciu:

Normal and compact families of BMO- and BMO_loc-QC mappings, Math. Reports 2(51), 4(2000) in print and

BMO_loc-QC mappings between Klein surfaces, Libertas Math. 20(2000), 7-13.

Date received: May 31, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-50.