Milnor invariants and twisted Whitney towers
The authors continue their work on higher order intersection invariants and Whitney towers, presented in earlier papers cited as [\textit{J. Conant} et al., Geom. Topol. 16, No. 3, 1419--1479 (2012; Zbl 1257.57005)]; [ibid. 16, No. 1, 555--600 (2012; Zbl 1284.57007)]; [Proc. Natl. Acad. Sci. USA 108, No. 20, 8131--8138 (2011; Zbl 1256.57017)] and [\textit{R. Schneiderman} and \textit{P. Teichner}, Geom. Topol. Monogr. 7, 101--134 (2004; Zbl 1090.57013)] in the list of references. The set \({\mathbb L}\) of oriented classical links contains Milnor's filtration \( ... \subseteq {\mathbb M}_n \subseteq {\mathbb M}_{n-1} \subseteq ... \subseteq {\mathbb M}_1 \subseteq {\mathbb L} \,\) where \({\mathbb M}_n\) contains all links whose \((k+1)\)-variable Milnor invariants are defined and vanish for all \(k \leq n\). Their twisted Whitney tower filtration \( ... \subseteq {\mathbb W}^{\infty}_n \subseteq {\mathbb W}^{\text{twist}}_{n-1} \subseteq ... \subseteq {\mathbb W}^{\text{twist}}_1 \subseteq {\mathbb L} \,\) is contained in Milnor's filtration, \({\mathbb W}^{\text{twist}}_n \subseteq {\mathbb M}_n\), where \({\mathbb W}^{\text{twist}}_n\) consists of all links \(L\) that bound a twisted Whitney tower \(\mathcal{W}\) of order \(n\) in the \(4\)-ball. They prove, moreover, for such an \(L\), the order \(k\) Milnor invariants \(\mu_k(L)\) vanish for \(k < n\) and \(\mu_n(L)\) can be computed from their intersection invariant \(\tau^{\text{twist}}_n(\mathcal{W})\) of the twisted Whitney tower \(\mathcal{W}\) via an equation \(\mu_n(L) = \eta_n \circ \tau^{\text{twist}}_n(\mathcal{W}) \in \mathsf{D}_n\). Here, \(\mathsf{D}_n\) is a free abelian group of known finite rank, and if \(W^{\text{twist}}_n\) is \({\mathbb W}^{\text{twist}}_n\) equipped with the band connected sum, then \(\mu_n: W^{\text{twist}}_n \rightarrow \mathsf{D}_n\) is found out to be an epimorphism which is bijective for all \(n \not\equiv 2 \mod 4\). For \(n = 4k-2\), higher order Arf invariants \(\mathrm{Arf}_k\) come in. They live on \(\ker \mu_{4k-2}\); \(\mathrm{Arf}_1\) is given by the classical Arf invariant of the link components (\(\tau^{\text{twist}}_2(\mathcal{W})\) for a knot). Whether \(\mathrm{Arf}_k\) is non-trivial, is open for \(k>1\). A remarkable result is therefore that \(\mathrm{Arf}_k\) is trivial for all \(k \geq 2\) if \(\mathrm{Arf}_2\) is trivial. Improving upon a result of \textit{K. Igusa} and \textit{K. E. Orr} [Topology 40, No. 6, 1125--1166 (2001; Zbl 1002.57012)], the authors also show that a link \(L\) is geometrically \(k\)-slice if and only if it is \(k\)-slice and \(\mathrm{Arf}_n(L) = 0\) for all \(n \leq k\).NEWLINENEWLINE NEWLINEEditor's remark: As upper index in their filtration the authors use a `twist'-symbol. This symbol is replaced by the word `\(\text{twist}\)' here.
scientific article; zbMATH DE number 6340746
scientific article