  We study surface representatives of homology classes of finite complexes
which minimize certain complexity measures, including its genus and Euler
characteristic. Our main result is that up to surgery at nullhomotopic curves
minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we
conclude that the minimizing problem is in general algorithmically undecidable,
but can be solved for 2-dimensional CAT(-1)-complexes.
