Equivariantly formal 2-torus actions of complexity one

In this paper we study a specific class of actions of a $2$-torus $\mathbb{Z}_2^k$ on manifolds, namely, the actions of complexity one in general position. We describe the orbit space of equivariantly formal $2$-torus actions of complexity one in general position and restricted complexity one actions in the case of small covers. It is observed that the orbit spaces of such actions are topological manifolds. If the action is equivariantly formal, we prove that the orbit space is a $\mathbb{Z}_2$-homology sphere. We study a particular subclass of these $2$-torus actions: restrictions of small covers to a subgroup of index 2 in general position. The subgroup of this form exists if and only if the small cover is orientable, and in this case we prove that the orbit space of a restricted $2$-torus action is homeomorphic to a sphere.