Curvature bounds of subsets in dimension two (Q6593657)

Let \(A\) \(\subset\) \(X\) be a closed, rectifiably connected subset with \(H_1(A) = 0\) in a two-dimensional contractible CAT(\(\kappa\)) space \(X\). The main result of the paper is that \(A\) is also a CAT(\(\kappa\)) space with respect to the induced intrinsic metric. This result holds in dimension one, as any one-dimensional CAT(\(\kappa\)) space is covered by a tree. However, the authors note that it fails in dimension at least three, since the complement of an open ball in \(\mathbb{R}^3\) is not non-positively curved and not aspherical.\N\NThe authors show also that if \(A \subset Y\) is an arbitrary subset in a two-dimensional non-positively curved space \(Y\), then every Lipschitz \(n\)-sphere in \(A\) with \(n \geq 2\) bounds a Lipschitz ball in \(A\).\N\NThe above result motivates the authors to make this \textit{conjecture}: If \(X\) is an aspherical space of topological dimension two, then any subset \(A\) of \(X\) is aspherical.