Classifying overlay structures of topological spaces
by
Vlasta Matijevic
Department of mathematics, University of Split, Croatia

The well-known classification theorem for covering mappings establishes a bijection between classes of equivalent s-sheeted pointed covering mappings f\colon X --> Y over a connected locally pathwise connected semi-locally 1-connected space Y and subgroups H of index s of the fundamental group \pi₁(Y, * ). In 1972 R.H. Fox [1] obtained a version of this result valid for arbitrary connected metric spaces Y. However, he had to replace covering mappings by newly introduced overlay structures and \pi₁(Y, * ) by its shape-theoretic analogue, the fundamental progroup \pi₁(Y, * ). By definiton, an s-sheeted overlay structure over a connected space Y is a fibre bundle, i.e. an equivalence class of coordinate bundles (X, Y, f, S, G), where the fibre S is discrete and of cardinality s, the trasformation group G is the symmetric group \Sigma(s) and the coordinate transformations are constant functions. If the base space Y is locally connected and paracompact or if s is finite, s-sheeted (indecomposable) overlay structures coincide with s-sheeted covering mappings (where the covering space X is connected).

Using ANR-resolutions of (Y, * ), the fact that overlays are pull-backs of covering mappings of ANR's, as well as the classical classification theorem, we generalize Fox's result to a classification theorem for overlay structures over arbitrary connected topological spaces [3]:

THEOREM. There exists a bijection between the set of all equivalent classes of s-sheeted indecomposable pointed overlay structures over a pointed connected topological space (Y, * ) and the set of all subprogroups of index s of the fundamental progroup \pi₁(Y, * ).

As in the classical case, the unpointed version of the theorem uses conjugacy classes of subprogroups of index s of the fundamental progroup \pi₁(Y, * ). Both results can also be stated in terms of transitive representations of the fundamental progroup \pi₁(Y, * ) in the symmetric group \Sigma(s).

Using the Cech covering approach, L.J. Hernández-Paricio [2] obtained a version of the classification theorem, which even allows non-connected base spaces and therefore, it uses fundamental progrupoids.

References

[ \text1] R.H. Fox, On shape, Fund. Math. 74 (1972), 47-71.

[ \text2] L.J. Hernández-Paricio,  Fundamental pro-grupoids and covering projections, Fund. Math. 156 (1998), 1-31.

[ \text3] S. Mardesi\'c and V. Matijevi\'c, Classifying overlay structures of topological spaces, Topology and Appl. (to appear).

Date received: March 8, 2000

Copyright © 2000 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 # cadz-39.