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Decorated marked surfaces: spherical twists versus braid twists

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Abstract

We are interested in the 3-Calabi-Yau categories \({\mathcal {D}}\) arising from quivers with potential associated to a triangulated marked surface \(\mathbf {S}\) (without punctures). We prove that the spherical twist group \(\mathrm{ST}\) of \({\mathcal {D}}\) is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface \({\mathbf {S}}_\bigtriangleup \). Here \({\mathbf {S}}_\bigtriangleup \) is the surface obtained from \(\mathbf {S}\) by decorating with a set of points, where the number of points equals the number of triangles in any triangulations of \(\mathbf {S}\). For instance, when \(\mathbf {S}\) is an annulus, the result implies that the corresponding spaces of stability conditions on \({\mathcal {D}}\) are contractible.

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Acknowledgments

This work was inspired during joint working with Alastair King on the twin paper [15], which deals with punctured marked surfaces. I would like to thank my collaborators mentioned above, as well as Tom Bridgeland, Ivan Smith, Dong Yang, Idun Reiten and Bernhard Keller for inspiring conversations.

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  1. Institutt for Matematiske Fag, NTNU, 7491, Trondheim, Norway

    Yu Qiu

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  1. Yu Qiu
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Correspondence to Yu Qiu.

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Qiu, Y. Decorated marked surfaces: spherical twists versus braid twists. Math. Ann. 365, 595–633 (2016). https://doi.org/10.1007/s00208-015-1339-0

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  • DOI: https://doi.org/10.1007/s00208-015-1339-0

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