Brieskorn spheres, cyclic group actions and the Milnor conjecture
type: Item
instance of: scholarly article
title: Brieskorn spheres, cyclic group actions and the Milnor conjecture
zbMATH Open document ID: 1546.57005
author: David Baraglia
author: Pedram Hekmati
published in: Journal of Topology
publication date: 2024-07-01T00:00:00Z
review text: The article further develops the Seiberg-Witten-Floer cohomology theory as first developed in [\textit{D. Baraglia} and \textit{P. Hekmati}, Algebr. Geom. Topol. 24, No. 1, 493--554 (2024; Zbl 1545.57013)] and applies it to Brieskorn homology spheres. Applications include a new proof of the Milnor conjecture and obstructions to extending group actions over a bounding 4-manifold. Let \(a_{1},\dots,a_{r}>1\) be pairwise coprime positive integers and let \(Y=K(a_{1},\dots,a_{r})\) be the Brieskorn homology sphere. \(Y\) is an integral homology 3-sphere and is also a Seifert 3-manifold, so there is an \(S^{1}\)-action on \(Y\). A \(\mathbb{Z}_{p}\)-action on \(Y\) is defined by restricting the circle action. The main results are obtained by considering the equivariant Seiberg-Witten-Floer cohomology of \(Y\) with respect to such \(\mathbb{Z}_{p}\)-actions. Let \(p>1\) be a prime and let \(Y=\Sigma_{p}(K)\) be the cyclic \(p\)-fold cover of \(S^{3}\) branched over a knot \(K\). The equivariant Seiberg-Witten-Floer cohomology of \(Y\) is applied to define knot concordance invariants \(\theta^{(p)}(K)\). The invariants are lower bounds for the slice genus \(g_{4}(K)\); indeed, \(g_{4}(K)\ge \theta^{(p)}(K)\) for all primes \(p\). The article considers the case where \(K=T_{a,b}\) is an \((a,b)\)-torus knot. Let \(c\) be a prime not dividing \(ab\), then \(\Sigma_{c}(T_{a,b})=K(a,b,c)\). The \(\mathbb{Z}_{c}\)-action resulting from the branched covering construction coincides with the restriction to \(\mathbb{Z}_{c}\) of the Seifert circle action.\N\NTheorem 1. Let \(a, b>1\) be coprimes and let \(c\) be a prime that does not divide the product \(ab\). Then \(\theta^{(c)}(T_{a,b})=\frac{1}{2}(a-1)(b-1)\).\N\NTherefore, \(g_{4}(K)\ge \frac{1}{2}(a-1)(b-1)\). The reverse is easy because the Milnor fiber of the singularity \(x^{a}=y^{b}\) has genus \(\frac{1}{2}(a-1)(b-1)\). So the Milnor conjecture \(g_{4}(K)= \frac{1}{2}(a-1)(b-1)\) follows as a consequence.\N\NThe equivaria…
reviewed by: Celso M. Doria
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 57K41
Mathematics Subject Classification ID: 57K31
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 7873633
zbMATH Keywords: Seiberg-Witten
zbMATH Keywords: Floer
zbMATH Keywords: Brieskorn spheres
MaRDI profile type: Publication
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DOI: 10.1112/TOPO.12339
description: scientific article; zbMATH DE number 7873633