Stratification of \(\mathrm{SU}(r)\)-character varieties of twisted Hopf links. (English) Zbl 07915461
Gothen, Peter (ed.) et al., Moduli spaces and vector bundles – new trends. VBAC 2022 conference in honor of Peter Newstead’s 80th birthday, University of Warwick, Coventry, United kingdom, July 25–29, 2022. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 803, 257-278 (2024).
Summary: We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with \(n\) twists, namely \(\Gamma_n=\langle x,y\mid[x^n,y]=1 \rangle\) into the group \(\mathrm{SU}(r)\). For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case \(r=2\), we provide a complete geometric description of the character variety, proving that this \(\mathrm{SU}(2)\)-character variety is a deformation retract of the larger \(\mathrm{SL}(2,\mathbb{C})\)-character variety, as conjectured by Florentino and Lawton. In the case \(r=3\), we also describe different strata of the \(\mathrm{SU}(3)\)-character variety according to the semi-simple type of the representation.
For the entire collection see [Zbl 07913515].
For the entire collection see [Zbl 07913515].
MSC:
| 14M35 | Character varieties |
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 53Cxx | Global differential geometry |
References:
| [1] | Biswas, Indranil, The topology of parabolic character varieties of free groups, Geom. Dedicata, 143-159, 2014 · Zbl 1295.14042 · doi:10.1007/s10711-012-9822-1 |
| [2] | Boden, Hans U., Representations of orbifold groups and parabolic bundles, Comment. Math. Helv., 389-447, 1991 · Zbl 0758.57013 · doi:10.1007/BF02566657 |
| [3] | Burde, Gerhard, Links and Seifert fiber spaces, Duke Math. J., 89-93, 1970 · Zbl 0195.54003 |
| [4] | Casimiro, Ana, On the homotopy type of free group character varieties, Bol. Soc. Port. Mat., 53-57, 2016 |
| [5] | H. Chen and T. Yu, The \(\operatorname SL(2, C)\)-character variety of the Borromean link, 2202.07429. |
| [6] | Florentino, C., Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone, Bull. Sci. Math., 84-101, 2019 · Zbl 1470.14065 · doi:10.1016/j.bulsci.2018.10.002 |
| [7] | Florentino, Carlos, The topology of moduli spaces of free group representations, Math. Ann., 453-489, 2009 · Zbl 1200.14093 · doi:10.1007/s00208-009-0362-4 |
| [8] | Florentino, Carlos, In the tradition of Ahlfors-Bers. VI. Character varieties and moduli of quiver representations, Contemp. Math., 9-38, 2013, Amer. Math. Soc., Providence, RI · Zbl 1325.16010 · doi:10.1090/conm/590/11720 |
| [9] | Florentino, Carlos, Topology of character varieties of Abelian groups, Topology Appl., 32-58, 2014 · Zbl 1300.14045 · doi:10.1016/j.topol.2014.05.009 |
| [10] | Florentino, Carlos, Homotopy groups of free group character varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 143-185, 2017 · Zbl 1403.14011 |
| [11] | C. Florentino and S. Lawton, Flawed groups and the topology of character varieties, 2012.08481. |
| [12] | Furuta, Mikio, Seifert fibred homology \(3\)-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math., 38-102, 1992 · Zbl 0769.58009 · doi:10.1016/0001-8708(92)90051-L |
| [13] | Gonz\'{a}lez-Prieto, \'{A}ngel, Geometry of SU(3)-character varieties of torus knots, Topology Appl., part A, Paper No. 108586 pp., 2023 · Zbl 1521.14092 · doi:10.1016/j.topol.2023.108586 |
| [14] | Gonz\'{a}lez-Prieto, \'{A}ngel, Motive of the \(\operatorname{SL}_4\)-character variety of torus knots, J. Algebra, 852-895, 2022 · Zbl 1502.14133 · doi:10.1016/j.jalgebra.2022.06.008 |
| [15] | Gonz\'{a}lez-Prieto, \'{A}ngel, Representation varieties of twisted Hopf links, Mediterr. J. Math., Paper No. 89, 30 pp., 2023 · Zbl 1516.57032 · doi:10.1007/s00009-023-02300-w |
| [16] | Griffiths, Phillip, Principles of algebraic geometry, Pure and Applied Mathematics, xii+813 pp., 1978, Wiley-Interscience [John Wiley & Sons], New York · Zbl 0408.14001 |
| [17] | Lawton, Sean, Character varieties of free groups are Gorenstein but not always factorial, J. Algebra, 278-293, 2016 · Zbl 1354.14021 · doi:10.1016/j.jalgebra.2016.01.044 |
| [18] | Mart\'{\i }nez-Mart\'{\i }nez, Javier, The \(SU(2)\)-character varieties of torus knots, Rocky Mountain J. Math., 583-602, 2015 · Zbl 1349.14048 · doi:10.1216/RMJ-2015-45-2-583 |
| [19] | Mehta, V. B., Moduli of vector bundles on curves with parabolic structures, Math. Ann., 205-239, 1980 · Zbl 0454.14006 · doi:10.1007/BF01420526 |
| [20] | Mu\~{n}oz, Vicente, The \(\operatorname{SL}(2,\mathbb{C})\)-character varieties of torus knots, Rev. Mat. Complut., 489-497, 2009 · Zbl 1182.14007 · doi:10.5209/rev\_REMA.2009.v22.n2.16290 |
| [21] | Nasatyr, Ben, Orbifold Riemann surfaces and the Yang-Mills-Higgs equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 595-643, 1995 · Zbl 0867.58009 |
| [22] | Newstead, P. E., Topological properties of some spaces of stable bundles, Topology, 241-262, 1967 · Zbl 0201.23401 · doi:10.1016/0040-9383(67)90037-7 |
| [23] | Newstead, P. E., Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vi+183 pp., 1978, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi · Zbl 1277.14001 |
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