Monodromy representations of branched covering maps between manifolds

Abstract

A monodromy representation represents a branched covering map between manifolds as a factor of an orbit map. This factorization arises from the monodromy of the branched covering map. We call the domain of the orbit map the monodromy space of the branched covering map. In this thesis a manifold is a connected second countable Hausdorff space that is locally homeomorphic to the Euclidean space. A branched covering map between manifolds is an open, continuous and discrete map. A branch point of a branched covering map is a point at which the map fails to be a local homeomorphism, and the branch set is the set of branch points. The first research question of this thesis considers new existence results of monodromy representations: In what extent does Bersteins and Edmonds construction of a monodromy representation generalize to branched covering maps between manifolds? In article [A] we first consider a class of branched covering maps between manifolds that is natural for the construction. Then we introduce a characterization for the existence of a monodromy representation within this class of maps. The second research question concerns the properties of the monodromy space: What properties can we expect from the monodromy space as a topological space? The monodromy space is a locally connected Hausdorff space by construction. In article [A] we provide an example to show that the monodromy space is not in general a locally compact space. In article [C] we provide further examples to show that a monodromy space that is a locally compact space, is not in general a manifold, a locally contractible space or a cohomology manifold. The third research question concerns new applications of monodromy representations in the study of branched covering maps between manifolds: What consequences does the existence of a monodromy representation have on the branch set of a branched covering map between manifolds? The branch set has by Väisälä codimension more or equal to two. A classical conjecture of Church and Hemmingsen states that the codimension of the branch set is strictly two for a branched covering map from the three sphere to a three sphere. We show together with Pankka in article [B] two partial results in the direction of this conjecture.Berstein ja Edmonds osoittivat, että suljettujen kompaktien monistojen välisillä haarautuvilla peitekuvauksilla on monodromiaesitys. Monodromiaesitys esittää haarautuvan peitekuvauksen ryhmätoiminnasta syntyvän normalisaatiokuvauksen tekijänä. Tässä väitöstutkimuksessa yleistetään Bersteinin ja Edmondsin tulosta monodromiaesityksen olemassaolosta laajemmalle luokalle monistojen välisiä haarautuvia peitekuvauksia, sovelletaan monodromiaesityksen olemassaoloa haarautuvan peitekuvauksen ominaisuuksien tutkimiseen ja perehdytään monodromiaesitykseen liittyvän monodromia-avaruuden ominaisuuksiin