The Magnus representation and homology cobordism groups of homology cylinders (Q2786744)

Let \(X\) be a compact oriented \(k\)-dimensional manifold. A \textit{homology cylinder} \((M, i_+, i_-)\) over \(X\) consists of a compact oriented \((k+1)\)-dimensional manifold \(M\) with two embeddings \(i_+, i_-: X \hookrightarrow \partial M\) such that (i) \(i_+\) is orientation preserving and \(i_-\) is orientation reversing, (ii) \(\partial M = i_+(X) \cup i_-(X)\) and \(i_+(X) \cap i_-(X) = i_+(\partial X) = i_-(\partial X)\), (iii) \(i_+|_{\partial X} = i_-|_{\partial X}\), (iv) \(i_+\) and \(i_-\) induce isomorphisms on the homology groups. The set of all isomorphisms classes \(\mathcal{C}(X)\) of homology cylinders over \(X\) has natural product operation given by stacking \((M, i_+, i_-)\) over \((N, j_+, j_-)\) by the identification of \(i_-(x)\) with \(j_+(x)\) for each \(x \in X\). Two homology cylinders \((M, i_+, i_-)\) and \((N, j_+, j_-)\) over \(X\) are said to be \textit{homology cobordant} if there exists a compact oriented \((k+2)\)-dimensional manifold \(W\) such that (i) \(\partial W = M \cup (-N)\) with \(i_+(x) = j_+(x)\) and \(i_-(x) = j_-(x)\) for each \(x \in X\), (ii) the inclusions \(M \hookrightarrow W\) and \(N \hookrightarrow W\) induce isomorphisms on the homology groups. The group \(\mathcal{H}(X)\) is the quotient set of \(\mathcal{C}(X)\) with respect to homology cobordism, and is called the \textit{homology cobordism group} of homology cylinders over \(X\).NEWLINENEWLINEIn the first half of this paper (Sections 3, 4, 5), abelian quotients of \(\mathcal{H}(X)\) are investigated when \(k=2\). Let \(\Sigma_{g,1}\) be a compact connected oriented surface of genus \(g\) with one boundary component. Let \(\mathcal{M}_{g,1}\) denote the mapping class group of \(\Sigma_{g,1}\), \(\mathcal{C}_{g,1} = \mathcal{C}(\Sigma_{g,1})\) and \(\mathcal{H}_{g,1} = \mathcal{H}(\Sigma_{g,1})\). For a self-diffeomorphism \(\varphi\) of \(\Sigma_{g,1}\), we can construct a homology cylinder by \((\Sigma_{g,1}, \mathrm{id} \times 1, \varphi \times 0)\) and this construction gives a monoid homomorphism from \(\mathcal{M}_{g,1}\) to \(\mathcal{C}_{g,1}\). \textit{S. Garoufalidis} and \textit{J. Levine} [in: M. Lyubich (ed.) et al., Graphs and patterns in mathematics and theoretical physics. Proceedings of the conference dedicated to Dennis Sullivan's 60th birthday, June 14--21, 2001. Providence, RI: American Mathematical Society (AMS) Proceedings of Symposia in Pure Mathematics 73, 173--203 (2005; Zbl 1086.57013)] showed that the monoid homomorphism \(\mathcal{M}_{g,1} \to \mathcal{C}_{g,1}\) and the composition \(\mathcal{M}_{g,1} \to \mathcal{C}_{g,1} \to \mathcal{H}_{g,1}\) are injective. \textit{J. Harer} [Invent. Math. 72, 221--239 (1983; Zbl 0533.57003)] showed that the abelianization of \(\mathcal{M}_{g,1}\) is trivial except for a few low genus cases. Nevertheless, the abelianization of \(\mathcal{H}_{g,1}\) has not yet been determined. In this paper, by using the Magnus representation together with Cha-Friedl-Kim's reduction technique [\textit{J. C. Cha} et al., Compos. Math. 147, No. 3, 914--942 (2011; Zbl 1225.57003)], abelian quotients of \(\mathcal{H}_{g,1}\) are investigated. Let \(\mathcal{K}_H = \mathbb{Z}[H_1(\Sigma_{g,1})] (\mathbb{Z}[H_1(\Sigma_{g,1})]-\{0\})^{-1}\). The Magnus representation for \(\mathcal{C}_{g,1}\) is the map \(r : \mathcal{C}_{g,1} \to \mathrm{GL}(2g, \mathcal{K}_{H})\) which assigns to \(M = (M, i_+, i_-) \in \mathcal{C}_{g,1}\) the matrix \(r(M)\) representing the isomorphism \(i_- \circ (i_+)^{-1} : \mathcal{K}_{H}^{2g} \to \mathcal{K}_{H}^{2g} \). Since the relative complex \(C_*(M,i_+(\Sigma_{g,1}); \mathcal{K}_{H_1(M)})\) is acyclic, we can define the Reidemeister torsion for this complex. Let \(\tau(M) \in \mathcal{K}_H^{\times}/(\pm H)\) be the image of this Reidemeister torsion under the identification of \(H_1(M)\) with \(H_1(\Sigma_{g,1})\). We consider two maps \(\hat{r} : \mathcal{H}_{g,1} \mathop{\rightarrow}\limits^{r} \mathrm{GL}(2g,\mathcal{K}_H) \mathop{\rightarrow}\limits^{\det} \mathcal{K}_H^{\times} \to \mathcal{K}_H^{\times} /(\pm H)\), \(\tilde{r} : \mathcal{H}_{g,1} \mathop{\rightarrow}\limits^{\hat{r}} \mathcal{K}_{H}^{\times}/(\pm H) \to \mathcal{K}_H^{\times}/(\pm H \cdot A)\), where \(A = \{ f^{-1} \cdot \varphi(f) \mid f \in \mathcal{K}^{\times}_H, \varphi \in \mathrm{Sp}(2g, \mathbb{Z}) \}\). The main result of the first half of this paper is : (Theorem 5.3) (1) for \((M, i_+, i_-) \in \mathcal{C}_{g,1}\), \(\hat{r}(M) = \overline{\tau(M)}(\tau(M))^{-1}\), (2) for \(g \geq 1\), \(\tilde{r}: \mathcal{H}_{g,1} \to \mathcal{K}_X^{\times}/ (\pm H \cdot A)\) is trivial, (3) for \(g \geq 2\), the image of the homomorphism \(\hat{r} | _{\mathcal{IH}_{g,1}}\) is isomorphic to \(\mathbb{Z}^{\infty}\), where \(\mathcal{IH}_{g,1}\) is the kernel of a group homomorphism \(\sigma : \mathcal{H}_{g,1} \to \mathrm{Sp}(2g,\mathbb{Z})\) defined by assigning to \((M, i_+, i_-)\) the isomorphism \(i_+^{-1} \circ i_-\) of \(H_1(\Sigma_{g,1})\).NEWLINENEWLINEIn the last half of this paper (Section 6, 7, 8), the argument of the first half of the paper is applied to the higher dimensional cases and it is shown that the determinant of the Magnus representation works well for them. Let \(X_n^k = \#_n (S^1 \times S^{k-1})\). The Magnus representation \(r : \mathcal{C}(X_n^k - \mathrm{Int} D^k) \to \mathrm{GL}(n,\mathcal{K}_{H_1})\) is defined by the same procedure as in the case where \(k=2\). We consider a homomorphism \(\tilde{r} : \mathcal{H}(X_n^k - \mathrm{Int} D^k) \mathop{\rightarrow}\limits^{r} \mathrm{GL}(n,\mathcal{K}_{H_1}) \mathop{\rightarrow}\limits^{\det} \mathcal{K}_{H_1}^{\times} \to \mathcal{K}_{H_1}^{\times} / (\pm H_1 \cdot A') \cong \mathbb{Z}^{\infty}\). The main result of the last half of this paper is: (Theorem 6.1) for any \(k \geq 3\) and \(n \geq 2\), (1) the image of \(\tilde{r}\) is an infinitely generated subgroup of \(\mathbb{Z}^{\infty}\), (2) \(\tilde{r}\) factors through \(\mathcal{H}(X_n^k)\).