UCLA

The work in this dissertation is centered around the study of quasiregularly elliptic manifolds. These are manifolds that admit quasiregular maps from Euclidean space. The research of quasiregular maps is motivated by the pursuit of extending theorems from complex analysis and conformal geometry to higher dimensional settings.

We first provide a new proof for the Rickman-Picard theorem, which states that a non-constant quasiregular map from Euclidean space to a sphere may omit a bounded number of points depending on the dilatation of the map.

We next show that a closed, connected and orientable Riemannian manifold that is quasiregularly elliptic must have bounded dimension of the cohomology independent of the distortion of the map. The bound for the dimension is sharp and proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a simply connected manifold that does not admit a quasiregular mapping from Euclidean space. The result shown gives an affirmative answer to this question.

Lastly, we study the behavior of branched covers whose image of their branch set is contained in a simplicial complex. The image of the branch set of a piecewise linear branched cover between piecewise linear manifolds is a simplicial complex. We demonstrate that the reverse implication also holds. A branched cover from a sphere to a sphere with the image of the branch set contained in a codimension two simplicial complex is equivalent up to homeomorphism to a PL mapping. This extends a result by Martio and Srebro in the three dimensional setting.