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.
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Notes
An almost free action is an action with finite stabilizers. This situation may occur if disconnected stabilizers are allowed for the original action; however, in this case, we take coefficients in \(\mathbb {Q}\). For an almost free action of T on X, we have \(H^*_T(X;\mathbb {Q})\cong H^*(X/T;\mathbb {Q})\).
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Acknowledgements
The authors thank the anonymous referees for the numerous valuable comments on the first and second versions of the paper, especially for paying our attention that orientability assumption is required in all statements of the paper and that the reference to Poincaré conjecture in dimension 4 was missing. We also appreciate the advice of the referee to rewrite some arguments in the language of homology with closed supports.
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Ayzenberg, A., Masuda, M. Orbit Spaces of Equivariantly Formal Torus Actions of Complexity One. Transformation Groups (2023). https://doi.org/10.1007/s00031-023-09822-3
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DOI: https://doi.org/10.1007/s00031-023-09822-3