We discuss a dimension-free deformation theory for Alexander maps and its applications.
In 1920, J. W. Alexander proved that every closed orientable PL (piecewise linear) n-manifold can be triangulated so that any two neighboring n-simplices are mapped to
the upper and the lower hemispheres of Sn, respectively. Such maps are called Alexander maps. Rickman introduced a powerful 2-dimensional deformation method for Alexander maps, in his celebrated proof (1985) of the sharpness of the Picard theorem in R3
The higher dimensional topological deformation leads to a Hopf degree theorem for Alexander maps, and a Berstein-Edmonds-type existence theorem for branched covers between manifolds with boundary. The geometrical deformation may be used to extend an example of Heinonen-Rickman on wildly branching quasiregular maps, from dimension 3 to 4.
(This is joint work with Pekka Pankka.)
