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A Note on Representation Variety of Abelian Groups and Reidemeister Torsion

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Functional Analysis in Interdisciplinary Applications—II (ICAAM 2018)

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Abstract

Let S and G denote respectively the \(2-\)torus and one of the Lie groups \(\mathrm {GL}(n,\mathbb {C}),\) \(\mathrm {SL}(n,\mathbb {C}),\) \(\mathrm {SO}(n,\mathbb {C}),\) \(\mathrm {Sp}(2n,\mathbb {C}).\) In the present article, we consider the smooth part of the representation variety \( \mathrm {Rep}(S,G) \) consisting of conjugacy classes of homomorphisms from fundamental group \(\pi _1(S)\) to G. We show the well definiteness of Reidemeister torsion for such representations. In addition, we establish a formula for computing the Reidemeister torsion of such representations in terms of the symplectic structure on \(\text {Rep}\left( S ,G\right) \) [51]. This symplectic form is analogous to Atiyah–Bott–Goldman symplectic form of higher genera for the Lie group G.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Arts, Duzce University, 81620, Duzce, Turkey

    Fatih Hezenci

  2. Department of Mathematics, Faculty of Science, Hacettepe University, 06800, Ankara, Turkey

    Yasar Sozen

Authors
  1. Fatih Hezenci
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  2. Yasar Sozen
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Correspondence to Fatih Hezenci .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Near East University, Mersin, Turkey

    Allaberen Ashyralyev

  2. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Tynysbek Sh. Kalmenov

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Michael V. Ruzhansky

  4. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Makhmud A. Sadybekov

  5. Department of Mathematics, Nazarbayev University, Astana, Kazakhstan

    Durvudkhan Suragan

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Hezenci, F., Sozen, Y. (2021). A Note on Representation Variety of Abelian Groups and Reidemeister Torsion. In: Ashyralyev, A., Kalmenov, T.S., Ruzhansky, M.V., Sadybekov, M.A., Suragan, D. (eds) Functional Analysis in Interdisciplinary Applications—II. ICAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 351. Springer, Cham. https://doi.org/10.1007/978-3-030-69292-6_13

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