Milnor invariants and twisted Whitney towers
type: Item
instance of: scholarly article
title: Milnor invariants and twisted Whitney towers
zbMATH Open document ID: 1312.57007
published in: Journal of Topology
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.0758
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 05Cxx
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 6340746
zbMATH Keywords: (framed) twisted Whitney tower
zbMATH Keywords: grope
zbMATH Keywords: framing
zbMATH Keywords: higher order Arf invariant
zbMATH Keywords: intersection number
zbMATH Keywords: Milnor invariants
zbMATH Keywords: geometrically \(k\)-slice
zbMATH Keywords: rooted tree
zbMATH Keywords: Whitney disk
zbMATH Keywords: filtration
MaRDI profile type: Publication
OpenAlex ID: W3106502862
arXiv ID: 1102.0758
DOI: 10.1112/JTOPOL/JTT025
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author: James Conant
author: Rob Schneiderman
author: Peter Teichner
review text: 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 \(\mat…
reviewed by: Dieter Erle
description: scientific article; zbMATH DE number 6340746