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Contractible stability spaces and faithful braid group actions

Yu Qiu Tsinghua University Jon Woolf University of Liverpool

Category Theory Representation Theory mathscidoc:1906.02001

Geometry & Topology, 22, 3701–3760, 2018
We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau–N category D(Gamma_N Q) associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group Br(Q) acts freely upon it by spherical twists, in particular that the spherical twist group Br(Gamma_N Q) is isomorphic to Br(Q). This generalises the result of Brav–Thomas for the N=2 case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.
Stability conditions, Calabi{Yau categories, spherical twists, braid groups
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@inproceedings{yu2018contractible,
  title={Contractible stability spaces and faithful braid group actions},
  author={Yu Qiu, and Jon Woolf},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625195402856079354},
  booktitle={Geometry & Topology},
  volume={22},
  pages={3701–3760},
  year={2018},
}
Yu Qiu, and Jon Woolf. Contractible stability spaces and faithful braid group actions. 2018. Vol. 22. In Geometry & Topology. pp.3701–3760. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625195402856079354.
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