Alexandrov spaces with integral current structure (Q2662949)

The main goal of the paper is to give the answer for the question, which Alexandrov spaces can be endowed with integral current structures in the following form: Theorem A: Let \((X, d)\) be a~closed, oriented, \(n\)-dimensional Alexandrov space with curvature bounded below by \(\kappa\). There exists an integral current structure \(T\) with weight equal to \(1\) defined on \(X\) such that \((X, d, T)\) is an \(n\)-dimensional integral current space. In [J. Topol. Anal. 12, No. 3, 819--839 (2020; Zbl 1446.49030)], \textit{N. Li} and \textit{R. Perales} proved under which assumptions the Gromov-Hausdorff and intrinsic flat limits of the sequence \((X_j, d_j, T_j)\) agree. Combining this fact with Theorem A leads to the following result: Theorem B: Let \(X_i\) be closed oriented \(n\)-dimensional Alexandrov spaces with curvature bounded below by \(\kappa\). Suppose further that \(\mathrm{diam}(X_i)\leq D\) and the sequence is non-collapsing. Then the \(X_i\) can be made into \(n\)-dimensional integral current spaces in a way that a subsequence converges in the intrinsic flat and Gromov-Hausdorff sense to the same metric space.