The 0-concordance monoid admits an infinite linearly independent set

Under the relation of \(0\)-concordance, the set of knotted 2-spheres in \(S^4\) forms a commutative monoid \(\mathcal{M}_0\) with the operation of connected sum. Sunukjian has recently shown that \(\mathcal{M}_0\) contains a submonoid isomorphic to \(\mathbb{Z}^{\ge 0}\). In this note, we show that...

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Bibliographic Details
Published in:arXiv.org 2023-09
Main Authors: Dai, Irving, Miller, Maggie
Format: Article
Language:English
Subjects:
Homology
Monoids
Online Access:Get full text
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Summary:Under the relation of \(0\)-concordance, the set of knotted 2-spheres in \(S^4\) forms a commutative monoid \(\mathcal{M}_0\) with the operation of connected sum. Sunukjian has recently shown that \(\mathcal{M}_0\) contains a submonoid isomorphic to \(\mathbb{Z}^{\ge 0}\). In this note, we show that \(\mathcal{M}_0\) contains a submonoid isomorphic to \((\mathbb{Z}^{\ge 0})^\infty\). Our argument relates the \(0\)-concordance monoid to linear independence of certain Seifert solids in the (spin) rational homology cobordism group.
ISSN:2331-8422