Finiteness and realization theorems for Alexandrov spaces with bounded curvature

Abstract

Every closed Alexandrov space with a lower and upper curvature bound (in the triangle comparison sense) is a space of bounded curvature (in the sense of Berestovskii and Nikolaev). These spaces are topological manifolds and admit a canonical smooth structure such that, in its charts, the metric is induced by a Riemannian metric tensor with low regularity. In this note, we show that there are finitely many diffeomorphism types of closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound, provided $$2\le n\le 6$$. We also show that an analogous diffeomorphism finiteness result for such spaces does actually hold for general dimension n if, in addition, the second homotopy group is finite. Moreover, we prove that all closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound can be realized as quotients of finitely many closed, simply connected smooth manifolds with finite second homotopy group by some free torus action. These results extend well-known finiteness and realization theorems in Riemannian geometry to the more general setting of Alexandrov geometry. Indeed, our results may be simply viewed as concrete instances of the more general phenomenon that virtually any result that holds for a given class of closed Riemannian manifolds with bounded curvature and diameter may be extended to the Lipschitz closure of this class.