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The bipolar filtration of topologically slice knots. (English) Zbl 1477.57002

The so-called “bipolar” filtration of [T. D. Cochran et al., Geom. Topol. 17, No. 4, 2103–2162 (2013; Zbl 1282.57006)] combines two central themes in knot concordance, by including information about both “definiteness of the intersection form motivated from Donaldson’s work [...] (and) fundamental group information related to derived subgroups and the tower techniques of Casson and Freedman for 4-manifolds and work of Cochran, Orr, and Teichner on knot concordance.” Somewhat more precisely, a knot is called \(n\)-positive (respectively, \(n\)-negative) if its zero-surgery bounds a 4-manifold \(V\) with the same intersection form as some connected sum of \(\mathbb{CP}^2\)’s (respectively, \(\bar{\mathbb{CP}^2}\)’s), such that the standard generators are represented by disjointly embedded surfaces whose fundamental groups live in the \(n\)th derived subgroup of \(\pi_1(V)\), together with some technical conditions. An \(n\)-bipolar knot is one that is both \(n\)-positive and \(n\)-negative. This gives a filtration of the smooth concordance group of topologically slice knots by subgroups \(\mathcal{T}_n\), and almost all of the known concordance invariants are known to vanish even for knots in \(\mathcal{T}_1\). In the original work of [T. D. Cochran et al., Geom. Topol. 17, No. 4, 2103–2162 (2013; Zbl 1282.57006)], it was shown that the first two levels of this filtration are nontrivial. In this work, Cha and Kim show that \(\mathcal{T}_n/ \mathcal{T}_{n+1}\) has infinite rank for all \(n \geq 2\). They rely on a delicate juxtaposition of \(d\)-invariants associated to infinitely many branched covers together with Cheeger-Gromov \(\rho\)-invariants, and their examples come from the winding number zero satellite constructions that have been ubiquitous in the study of knot concordance since at least [T. D. Cochran et al., Ann. Math. (2) 157, No. 2, 433–519 (2003; Zbl 1044.57001)].

MSC:

57K10 Knot theory
57K40 General topology of 4-manifolds
57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
57N70 Cobordism and concordance in topological manifolds

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