Extended Alexander matrices of 3-manifolds. I (Q1069180)

Let \(M=T\cup T'\) be a genus g Heegaard splitting of a closed orientable 3-manifold M. With any system of meridians and longitudes \(a_ 1,b_ 1,\ldots,a_ g,b_ g\) of the handlebody T' (where \(a_ 1,b_ 1,\ldots,a_ g,b_ g\) are loops in \(\partial T'=\partial T\) having a common initial point m) and with any system of free generators \(x_ 1,\ldots,x_ g\) of the free group \(F=\pi_ 1(T,m)\) the authors associate ''an extended Alexander matrix'' \(\left( \begin{matrix} A\\ B\end{matrix} \right)\). Here A is the image of the \(g\times g\)-matrix of Fox derivatives \((\partial \alpha_ i/\partial x_ j)_{i,j=1,\ldots,g}\) under the natural ring homomorphism \({\mathbb{Z}}[F]\to {\mathbb{Z}}[H_ 1(M)]\) where \(\alpha_ i\) denotes the class of \(a_ i\) in F and \(\partial \alpha_ i/\partial x_ j\in {\mathbb{Z}}[F]\). B is defined similarly using \(b_ i\) instead of \(a_ i.\) The authors introduce in the set of 2g\(\times g\)-matrices over \({\mathbb{Z}}[H_ 1(M)]\) with \(g=0,1,..\). a suitable equivalence relation (including stabilization) and show that the equivalence class of \(\left( \begin{matrix} A\\ B\end{matrix} \right)\) depends only on M. This gives rise to a new invariant of 3-manifolds. It is stated that the invariant obtained distinguishes lens spaces up to homeomorphism. A proof of this statement and some further development are postponed to a forthcoming paper of the first author.