An intrinsic Arf invariant on a link and its surface-link analogue
type: Item
instance of: scholarly article
title: An intrinsic Arf invariant on a link and its surface-link analogue
zbMATH Open document ID: 1002.57031
DOI: 10.1016/S0166-8641(01)00122-5
author: Akio Kawauchi
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: A quadratic function on a finite Abelian group \(G\) is a function \(q:G\to \mathbb Q/\mathbb Z\) such that \(q(-x)=q(x)\) for any \(x\in G\) and the pairing \(\ell :G\times G\to \mathbb Q/\mathbb Z\), defined by \(\ell(x,y)=q(x+y)-q(x)-q(y)\), is a non-singular symmetric bilinear form, called the linking induced by \(q\). The Gaussian sum of \(q\) is the complex number given by \(GS(q)=\sum_{x\in G} \text{exp}(2\pi \sqrt{-1} q(x))\). It turns out that \(GS(q)\) is an invariant of \(q\), up to isomorphism. The author proves that for every quadratic function \(q:G\to \mathbb Q/\mathbb Z\) on a finite Abelian group \(G\), there is a unique modulo one rational number \(\sigma (q)\in \mathbb Q/\mathbb Z\) of order up to 8 (that is, \(8\sigma(q)=0\) in \(\mathbb Q/\mathbb Z\)) such that \(GS(q)=\sqrt {|G|} \text{exp}(2\pi \sqrt{-1} \sigma(q))\). This number \(\sigma(q)\) is called the linking signature of \(q\), and it is proved to be a natural generalization of the Arf invariant of a \(\mathbb Z_2\)-hyperbolic quadratic function. Then the paper is devoted to showing the topological meaning of the linking signature. In fact, the author constructs a \(\mathbb Z_2\)-hyperbolic quadratic function invariant for every infinite cyclic covering \(\widetilde M\) of every (possibly non-orientable) compact 3-manifold \(M\), and a \(\mathbb Q/\mathbb Z\)-quadratic function invariant of every infinite cyclic covering \(\widetilde W\) of every compact oriented 4-manifold \(W\). The linking signature up to 2 (resp. up to 4) of the \(\mathbb Z_2\)-hyperbolic quadratic (resp. \(\mathbb Q/\mathbb Z\)-quadratic) function of \(\widetilde M\) (resp. \(\widetilde W\)) is called the intrinsic Arf invariant of \(\widetilde M\) (resp. \(\widetilde W\)), and denoted by \(\alpha(\widetilde M)\) (resp. \(\sigma(\widetilde W)\)). It is shown that these invariants become invariants of arbitrary oriented links in \(\mathbb S^3\) and surface-links in \(\mathbb S^4\), respectively. More precisely, the …
reviewed by: Fulvia Spaggiari
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 1787528
zbMATH Keywords: infinite cyclic covering
zbMATH Keywords: quadratic function
zbMATH Keywords: linking signature
zbMATH Keywords: 3-manifold
zbMATH Keywords: 4-manifold
zbMATH Keywords: link
zbMATH Keywords: surface link
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787528