Contractible flow of stability conditions via global dimension function

We introduce an analytic method that uses the global dimension function gldim to produce contractible flows on the space Stab $\mathcal{D}$ of stability conditions on a triangulated category $\mathcal{D}$ . In the case when $\mathcal{D}=\mathcal{D}\left(\mathbf{S}^\lambda\right)$ is the topological Fukaya category of a graded surface $\mathbf{S}^\lambda$ , we show that gldim ${ }^{-1}(0, y)$ contracts to gldim ${ }^{-1}(0, x)$ for any $1 \leq x \leq y$ , provided $(x, y)$ does not contain 'critical' values $\left\{1+w_\theta / m_\theta \mid w_\theta \geq 0, \partial \in \partial \mathbf{S}^\lambda\right\}$ , where the pair ( $m_\theta, w_\theta$ ) consists of the number $m_\theta$ of marked points and the winding number $w_\theta$ associated to a boundary component $\theta$ of $\mathbf{S}^\lambda$ . One consequence is that the global dimension of $\mathcal{D}\left(\mathbf{S}^\lambda\right)$ 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.