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Let a compact torus $T=T^{n-1}$ act on an orientable smooth compact manifold$X=X^{2n}$ effectively, with nonempty finite set of fixed points, and supposethat stabilizers of all points are connected. If $H^{odd}(X)=0$ and the weightsof tangent representation at each fixed point are in general position, we provethat the orbit space $Q=X/T$ is a homology $(n+1)$-sphere. If, in addition,$\pi_1(X)=0$, then $Q$ is homeomorphic to $S^{n+1}$. We introduce the notion of$j$-generality of tangent weights of torus action. For any action of $T^k$ on$X^{2n}$ with isolated fixed points and $H^{odd}(X)=0$, we prove that$j$-generality of weights implies $(j+1)$-acyclicity of the orbit space $Q$.This statement generalizes several known results for actions of complexity zeroand one. In complexity one, we give a criterion of equivariant formality interms of the orbit space. In this case, we give a formula expressing Bettinumbers of a manifold in terms of certain combinatorial structure that sits inthe orbit space.

Orbit spaces of equivariantly formal torus actions of complexity one