The dual boundary complex of the $SL_2$ character variety of a punctured sphere

Carlos Simpson

doi:10.5802/afst.1496

The dual boundary complex of the

S L_{2}

character variety of a punctured sphere

Carlos Simpson ¹

¹ CNRS, Laboratoire JAD, UMR 7351, Université Nice Sophia Antipolis, 06108 Nice Cedex 2, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 317-361.

Abstract
Abstract

Suppose $C_{1}, ..., C_{k}$ are generic conjugacy classes in $S L_{2} (ℂ)$ . Consider the character variety of local systems on $ℙ^{1} - {y_{1}, ..., y_{k}}$ whose monodromy transformations around the punctures $y_{i}$ are in the respective conjugacy classes $C_{i}$ . We show that the dual boundary complex of this character variety is homotopy equivalent to a sphere of dimension $2 (k - 3) - 1 .$

Soient $C_{1}, ..., C_{k}$ des classes de conjugaison génériques dans $S L_{2} (ℂ)$ . On considère la variété de caractères des systèmes locaux sur $ℙ^{1} - {y_{1}, ..., y_{k}}$ dont les transformations de monodromie autour des $y_{i}$ sont dans les classes de conjugaison $C_{i}$ respectives. On montre que le complexe dual du bord de cette variété est équivalent par homotopie à un sphère de dimension $2 (k - 3) - 1 .$

Published online: 2016-07-11

MR Zbl

DOI: 10.5802/afst.1496

Author's affiliations:

Carlos Simpson ¹

¹ CNRS, Laboratoire JAD, UMR 7351, Université Nice Sophia Antipolis, 06108 Nice Cedex 2, France

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     title = {The dual boundary complex of the $SL_2$ character variety of a punctured sphere},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {317--361},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
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Carlos Simpson. The dual boundary complex of the $SL_2$ character variety of a punctured sphere. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 317-361. doi : 10.5802/afst.1496. https://afst.centre-mersenne.org/articles/10.5802/afst.1496/

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