UV^k-mappings on homology manifolds

J. Bryant, S. Ferry, W. Mio

We prove control improvement theorems for mappings from a compact homology manifold of dimension greater than 4 with the disjoint disks property (DDP) to a polyhedron. The methods used apply to any euclidean neighborhood retract having sufficient general position properties, so the essential results are presented in this general setting. Taking limits, we show that mappings with sufficient controlled connectivity can be deformed to a mapping with arbitrarily fine connectivity below middle and adjacent dimensions. In particular, this extends a deformation result of Bestvina and Walsh for mappings with manifold domains to mappings defined on homology manifolds with the DDP.