An intrinsic Arf invariant on a link and its surface-link analogue
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 intrinsic Arf invariant of an oriented link \(L\) in \(\mathbb S^3\) is defined as \(\alpha(L):=\alpha(\widetilde E_L)\) for the unique infinite cyclic connected covering \(\widetilde E_L\) of the compact exterior \(E_L=\text{cl} (\mathbb S^3 \smallsetminus L)\) of \(L\) in \(\mathbb S^3\). The linking signature \(\sigma(F)\) of an oriented surface-link \(F\) in \(\mathbb S^4\) is defined as \(\sigma(F)=\sigma(\widetilde E_F)\) for the unique infinite cyclic connected covering \(\widetilde E_F\) of the compact exterior \(E_F=\text{cl}(\mathbb S^4 \smallsetminus F)\) of \(F\) in \(\mathbb S^4\). Several examples, computations and properties on these invariants complete this nice and interesting paper.
scientific article; zbMATH DE number 1787528
scientific article