Discrete derived categories II: the silting pairs CW complex and the stability manifold

Discrete derived categories were studied initially by Vossieck ['The algebras with discrete derived category', J. Algebra 243 (2001) 168-176] and later by BobiA"ski, Gei and SkowroA"ski ['Classification of discrete derived categories', Cent. Eur. J. Math. 2 (2004) 19-49]. In this article, we define the CW complex of silting pairs for a triangulated category and show that it is contractible in the case of discrete derived categories. We provide an explicit embedding from the silting CW complex into the stability manifold. By work of Qiu and Woolf ['Contractible stability spaces and faithful braid group actions', Preprint, 2014, arXiv:1407.5986], there is a deformation retract of the stability manifold onto the silting pairs CW complex. We obtain that the space of stability conditions of discrete derived categories is contractible.