Topological minimal genus and $L^2$-signatures

TL;DR

This paper introduces new lower bounds for the minimal genus of surfaces in 4-manifolds using von Neumann-Cheeger-Gromov invariants, with applications to knot slice genus, revealing cases where traditional invariants fail.

Contribution

It provides novel lower bounds on surface genus in 4-manifolds via $ ho$-invariants, improving understanding of knot slice genus and identifying cases where previous invariants are ineffective.

Findings

New lower bounds for minimal genus using $ ho$-invariants

Examples of knots with arbitrarily large slice genus where bounds are sharp

Previous invariants vanish in these cases, highlighting the bounds' effectiveness

Abstract

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.