Plan's theorem for links
type: Item
instance of: scholarly article
title: Plan's theorem for links
zbMATH Open document ID: 0699.57003
DOI: 10.1016/0166-8641(90)90041-Y
author: Pablo Del Val
author: Claude Weber
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Consider the n-fold cyclic branched covering \(K_ n\) of a link K in the 3-sphere \(S^ 3\) or, more generally, in an integral homology 3-sphere. When K is a knot, namely when K is connected, a theorem of Antonio Plans asserts that the first homology group \(H_ 1(K_ n)\) of \(K_ n\) is a direct double, namely splits as a direct sum \(H\oplus H\). This theorem fails when K is not connected, and the authors consider what remains of this result in this case. They show that, if n is odd, the torsion-free quotient and the p-torsion subgroups of \(H_ 1(K_ n)\) are direct doubles for all primes p not dividing n. For n even, there is a natural map \(\pi: K_ n\to K_ 2\), inducing a homomorphism \(\pi_*: H_ 1(K_ n)\to H_ 1(K_ n)\). The authors then show that the torsion-free quotient and the p-torsion groups of the kernel of \(\pi_*\) are direct doubles for all primes not dividing n/2.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 4147102
zbMATH Keywords: branched cyclic covering
zbMATH Keywords: transfer homomorphism
zbMATH Keywords: alternate form
zbMATH Keywords: direct double
zbMATH Keywords: n-fold cyclic branched covering of a link
zbMATH Keywords: first homology group
zbMATH Keywords: torsion-free quotient
zbMATH Keywords: p-torsion subgroups
reviewed by: Francis Bonahon
MaRDI profile type: Publication
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full work available at URL: 0166-8641(90)90041-y
OpenAlex ID: W2157911841
description: scientific article; zbMATH DE number 4147102