Topological minimal genus and $L^2$-signatures

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $\rho$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples with arbitrarily large slice genus for which our lower bound is optimal but all previously known invariants vanish.