#### Volume 22, issue 6 (2018)

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Contractible stability spaces and faithful braid group actions

### Yu Qiu and Jon Woolf

Geometry & Topology 22 (2018) 3701–3760
##### Abstract

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 $\mathsc{D}\left({\Gamma }_{N}Q\right)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $Br\left(Q\right)$ acts freely upon it by spherical twists, in particular that the spherical twist group $Br\left({\Gamma }_{N}Q\right)$ is isomorphic to $Br\left(Q\right)$. 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.

##### Keywords
stability conditions, Calabi–Yau categories, spherical twists, braid groups
##### Mathematical Subject Classification 2010
Primary: 18E30, 20F36
Secondary: 13F60, 32Q55
##### Publication
Received: 7 November 2017
Revised: 8 February 2018
Accepted: 13 March 2018
Published: 23 September 2018
Proposed: Richard Thomas
Seconded: Jim Bryan, Frances Kirwan
##### Authors
 Yu Qiu Yau Mathematical Sciences Center Tsinghua University Beijing, China Jon Woolf Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom