Contractible flow of stability conditions via global dimension function

Abstract

We introduce an analytic method that uses the global dimension function $\operatorname{gldim}$ to produce contractible flows on the space $\operatorname{Stab}\mathcal{D}$ of stability conditions on a triangulated category $\mathcal{D}$. In the case when $\mathcal{D}=\mathcal{D}(\mathbf{S}^\lambda)$ is the topological Fukaya category of a graded surface $\mathbf{S}^\lambda$, we show that $\operatorname{gldim}^{-1}(0,y)$ contracts to $\operatorname{gldim}^{-1}(0,x)$ for any $1\le x\le y$, provided $(x,y)$ does not contain `critical' values $\{1+w_\partial/m_\partial \mid w_\partial\ge0, \partial\in\partial\mathbf{S}^\lambda\}$, where the pair $(m_\partial,w_\partial)$ consists of the number $m_\partial$ of marked points and the winding number $w_\partial$ associated to a boundary component $\partial$ of $\mathbf{S}^\lambda$. One consequence is that the global dimension of $\mathcal{D}(\mathbf{S}^\lambda)$ must be one of these critical values. Besides, we remove the assumptions in Kikuta-Ouchi-Takahashi's classification result on triangulated categories with global dimension less than 1.