Simply connected Alexandrov 4-manifolds with positive or nonnegative curvature and torus actions

Abstract

We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-manifolds with positive or nonnegative sectional curvature and an effective isometric action of a circle or a 2-torus also hold if we consider Alexandrov manifolds instead of Riemannian manifolds.