We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group G for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.
Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions at most $7$.