Orbit spaces of equivariantly formal torus actions of complexity one

Abstract: 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 suppose that stabilizers of all points are connected. If $H^{odd}(X)=0$ and the weights of tangent representation at each fixed point are in general position, we prove that 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 zero and one. In complexity one, we give a criterion of equivariant formality in terms of the orbit space. In this case, we give a formula expressing Betti numbers of a manifold in terms of certain combinatorial structure that sits in the orbit space.