Volume pinching theorems for CAT(1) spaces

We examine volume pinching problems of ${\rm CAT}(1)$ spaces. We characterize a class of compact geodesically complete ${\rm CAT}(1)$ spaces of small specific volume. We prove a sphere theorem for compact ${\rm CAT}(1)$ homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume growths under an upper curvature bound.