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Torsion and symplectic volume in Seifert manifolds. (Torsion et volume symplectique des variétés de Seifert.) (English. French summary) Zbl 1407.30023

Let \(\Sigma \) be a compact oriented surface with \(n\geq 1\) boundary components \(C_{i}\), \(1\leq i\leq n\), and let \(X\) be the Seifert manifold obtained by gluing \(n\) copies of \(D\times {{S}^{1}}\) to \(\Sigma \times {{S}^{1}}\), where \(D\subset \mathbb{C}\) is the standard closed disk. For \(G\) a compact connected Lie group with finite center, the authors provide a relationship between the Reidemeister density of the moduli space \({\mathcal{M}}\left( X \right)\) of representations of \({{\pi }_{1}}\left( X \right)\) in \(G\) and the Liouville measure of certain subsets of the moduli space \({\mathcal{M}}\left( \Sigma \right)\) of representations of \({{\pi }_{1}}\left( \Sigma \right)\) in \(G\).
In particular, for \(Z\left( G \right)\) the center and for \({\mathcal{C}}\left( G \right)\) the set of conjugacy classes of the group \(G\), let \[ {\mathcal{P}}=\left\{ \left( u,v \right)\in {\mathcal{C}}{{\left( G \right)}^{n}}\times Z\left( G \right)\left| \text{ for any }i,\,{{u}^{{{p}_{i}}}}={{v}^{{{q}_{i}}}} \right. \right\}, \] where \({{p}_{i}}\geq 1\) and \({{q}_{i}}\) are coprime integers. The subset \({{{\mathcal{M}}}^{0}}\left( X \right)\subset {\mathcal{M}}\left( X \right)\) of irreducible representations decomposes as \[ {{{\mathcal{M}}}^{0}}\left( X \right)=\bigcup\limits_{\left( u,v \right)\in {\mathcal{P}}}{{{{\mathcal{M}}}^{0}}\left( X,u,v \right)}, \] where \({{{\mathcal{M}}}^{0}}\left( X,u,v \right)=\left\{ \left[ \rho \right]\in {{{\mathcal{M}}}^{0}}\left( X \right)\left| \rho \left( {{S}^{1}} \right)\in v\text{ and }\rho \left( {{C}_{i}} \right)\in {{u}_{i}}\text{, for any }i \right. \right\}\). Moreover, the restriction map \({{R}_{u,v}}:{{{\mathcal{M}}}^{0}}\left( X,u,v \right) \to {{{\mathcal{M}}}^{0}}\left( \Sigma ,u \right)\) with \(\left[ \rho \right] \mapsto \left[ \rho \left| _{\Sigma } \right. \right]\) is shown to be a bijection, where \[ {{{\mathcal{M}}}^{0}}\left( \Sigma ,u \right)=\left\{ \left[ \rho \right]\in {{{\mathcal{M}}}^{0}}\left( \Sigma \right)\left| \text{ for any }i,\text{ }\rho \left( {{C}_{i}} \right)\in {{u}_{i}} \right. \right\}. \] The authors show that there is an isomorphism \(\psi :{{H}_{1}}\left( X,\text{Ad}\rho \right)\to {{H}_{2}}\left( X,\text{Ad}\rho \right)\), and for \(\Delta :{\mathcal{C}}\left( G \right)\to \mathbb{R}\) the function given by \(\Delta \left( u \right)={{\left| \det {{H}_{g}}\left( \text{A}{{\text{d}}_{g}}-\text{id} \right) \right|}^{\frac{1}{2}}}\), where \(g\) is any element in the conjugation class \(u\) and \({{H}_{g}}\) denotes the orthocomplement of \(\ker \left( \text{A}{{\text{d}}_{g}}-\text{id} \right)\), they prove that for any \(\rho \in {{{\mathcal{M}}}^{0}}\left( X \right)\) the Reidemeister torsion of \(\text{Ad}\rho \to X\) is given by \[ \tau \left( \text{Ad}\rho \right)=\prod\limits_{i=1}^{n}{\frac{p_{i}^{\dim{{V}_{i}}}}{{{\Delta }^{2}}\left( \rho {{\left( {{C}_{i}} \right)}^{{{r}_{i}}}} \right)}}\det \psi, \] where \({{r}_{i}}\) is any inverse of \({{q}_{i}}\) modulo \({{p}_{i}}\), and \({{V}_{i}}=\ker \left( \text{A}{{\text{d}}_{\rho \left( {{C}_{i}} \right)}}-\text{id} \right)\).
An application of this computation on the moduli space \({{{\mathcal{M}}}^{0}}\left( X \right)\) provides the main result of the article, which states that for any \(\left( u,v \right)\in {\mathcal{P}}\), the density \({{\mu }_{X}}\) of \({{{\mathcal{M}}}^{0}}\left( X \right)\) on \({{{\mathcal{M}}}^{0}}\left( X,u,v \right)\) is given by \[ {{\mu }_{X}}=\left( \prod\limits_{i=1}^{n}{\frac{\Delta \left( u_{i}^{{{r}_{i}}} \right)}{{{p}_{i}}^{\dim{{V}_{i}}/{2}}\;}} \right)R_{u,v}^{*}{{\mu }_{u}} \] for the aforementioned notation and for \({{\mu }_{u}}\) the canonical density (Liouville measure) of the symplectic manifold \({{{\mathcal{M}}}^{0}}\left( \Sigma ,u \right)\).
In the special case (abelian case) when the Euler number \(\chi =-\sum\limits_{i=1}^{n}{\frac{{{q}_{i}}}{{{p}_{i}}}}\) does not vanish, the authors compute the Reidemeister torsion of a Seifert manifold \(X\) to be given by \[ \tau \left( X \right)=\chi \prod\limits_{i=1}^{n}{{{p}_{i}}\left[ x \right]\otimes \det \psi \otimes \left[ X \right]}^{-1}, \] where \(\left[ x \right]\in {{H}_{0}}\left( X,\mathbb{R} \right)\) and \(\left[ X \right]\in {{H}_{3}}\left( X,\mathbb{R} \right)\) denotes the fundamental class.

MSC:

30F99 Riemann surfaces
53D30 Symplectic structures of moduli spaces
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
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