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Mayberry-Murasugi’s formula for links in homology 3-spheres. (English) Zbl 1064.57002

There is a famous formula by Fox for the order of the first homology group of a finite cyclic covering of a knot in the \(3\)-sphere from its Alexander polynomial. This has been generalized by J. P. Mayberry and K. Murasugi [Trans. Am. Math. Soc. 271, 143–173 (1982; Zbl 0487.57001)] for finite abelian coverings of links in the \(3\)-sphere. The paper under review gives a new proof of this formula using Franz-Reidemeister torsions which applies to links not only in the \(3\)-sphere but in homology \(3\)-spheres. The relationship between the Alexander polynomial and Franz-Reidemeister torsion was discovered by J. W. Milnor [Ann. Math. (2) 76, 137–147 (1962; Zbl 0108.36502)] and further developed by V. G. Turaev [Russ. Math. Surv. 41, No. 1, 119–182 (1986; Zbl 0602.57005)]. In particular, Turaev reproved Fox’s formula for knots in homology spheres, but not Mayberry-Murasugi’s. Let \(M\) be a closed homology \(3\)-sphere and let \(L\) be a link with \(\mu\) components, whose exterior is denoted by \(E(L)\). For a finite abelian group \(G\), the covering \(\hat{M}_{\pi}\) of \(M\) branched along \(L\) corresponds to the kernel of an epimorphism \(\pi:\pi_1(E(L))\to G\). The set of representations \(\xi:G\to \mathbb{C}^*\) from \(G\) to the nonzero complex numbers is denoted by \(\hat{G}\). Choose meridians \(m_1,\dots,m_\mu\). For a representation \(\xi\in \hat{G}\), let \(L_\xi\) denote the sublink consisting of the components with \(\xi(m_i)\neq 1\). Also, let \(\hat{G}^{(1)}\) denote the subset of representations \(\xi\in \hat{G}\) such that \(L_\xi\) is a knot. Finally, consider the product \(D\) of the moduli \(| \Delta_{L_{\xi}} (\xi(m_{i_1}), \xi(m_{i_2}),\dots,\xi(m_{i_k}))| \) over all \(\xi\in \hat{G}\), and the product \(R\) of the moduli \(| 1-\xi(m_i) | \) over all \(\xi\in \hat{G}^{(1)}\). Here, \(\Delta_{L_{\xi}}\) is the Alexander polynomial of \(L_{\xi}\). Then the order of \(H_1(\hat{M}_{\pi})\) is the product of \(| G| \) and \(D\) divided by \(R\). The proof is based on the interpretation of the order of \(H_1(\hat{M}_{\pi})\) as the torsion of the induced \(CW\)-complex \(\hat{K}\) for a \(CW\)-complex \(K\) with \(| K| =M\) and having \(L\) as a subset of the \(1\)-skeleton. There are two generalizations to coverings of homology \(3\)-spheres along graphs and to the case of higher-dimensional knots.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20K01 Finite abelian groups
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