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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 D(ΓNQ) 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(ΓNQ) 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.

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Keywords
stability conditions, Calabi–Yau categories, spherical twists, braid groups
Mathematical Subject Classification 2010
Primary: 18E30, 20F36
Secondary: 13F60, 32Q55
References
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