Contractible stability spaces and faithful braid group actions

2014/07/31 by Yu Qiu, Jon Woolf · 1 citation
Mathematics · #math.AG #math.RT #msc:14F05 #msc:16J99 #msc:18E30

paper · pdf · doi:10.2140/gt.2018.22.3701

published as Geom. Topol. 22 (2018) 3701-3760 · Final version, to appear in Geom. Topol

arxiv created 2018/04/02 · arxiv updated 2018/10/03

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(Γ_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(Γ_N Q) is isomorphic to Br(Q). This generalises Brav-Thomas' result 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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