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Topological minimal genus and \(L^2\)-signatures. (English) Zbl 1162.57016

The paper under review treats the problem of the minimal genus of a locally flatly embedded surface representing a given 2-dimensional homology class in a topological 4-manifold, in the case when the 4-manifold has boundary with nontrivial homology. It gives new lower bounds for the minimal genus for homology classes from the boundary, in terms of the von Neumann-Cheeger-Gromov \(\rho\)-invariants of the boundary. As an application, new lower bounds for the slice genus of a knot are given. In particular, for any positive integer \(g\), it is shown that there are infinitely many knots \(K\) with genus \(g\), topological slice genus and smooth slice genus also \(g\), but so that each \(K\) has the Seifert matrix of a slice knot, and vanishing Casson-Gordon invariants, Ozsváth-Szabó \(\tau\)-invariant and Rasmussen \(s\)-invariant.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N35 Embeddings and immersions in topological manifolds
57R95 Realizing cycles by submanifolds
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