On $3$-dimensional locally standard $T$-pseudomanifolds

Locally standard $T$-pseudomanifolds were introduced by the authors in a previous work. They are topological stratified pseudomanifolds equipped with torus actions. Their equivariant homeomorphism types are classified by characteristic data under the homotopy equivalence condition. In this paper, we show that this condition can be removed when the dimension is at most three. We also characterize those of dimension at most three that are topological manifolds, in terms of their orbit spaces.