The minimal genus of homology classes in a finite 2-complex (Q2172195)

For a fixed singular integral homology class \(\alpha\in H_2(X)\) of a CW-complex \(X\), maps of closed oriented (possibly disconnected) surfaces \(f\colon\Sigma\to X\) representing \(\alpha \) are studied. Let \(g(\Sigma )\) be the sum of \(g(\Sigma ')\) over all components \(\Sigma '\in\Sigma\) and let \[ \chi^-(\Sigma)=\sum_{\Sigma'}\max\{ 0,-\chi(\Sigma')\} . \] The main result is that, up to surgery at nullhomotopic curves, a \(\Sigma \) which minimizes \(g\) and \(\chi^-\) is homotopic to a cellwise covering of the 2-skeleton. From this the authors conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT\((-1)\) 2-complexes.