Orbit spaces of equivariantly formal torus actions of complexity one
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arXiv
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| Format: | Preprint |
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2019
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| author | Ayzenberg, Anton Masuda, Mikiya |
| author_facet | Ayzenberg, Anton Masuda, Mikiya |
| contents | 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, $π_1(X)=0$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_11696 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Orbit spaces of equivariantly formal torus actions of complexity one Ayzenberg, Anton Masuda, Mikiya Algebraic Topology Combinatorics K-Theory and Homology 57S25, 57N65, 55N91, 55N25, 06A07 (Primary) 55R20, 18G35, 18G10, 55N30, 06A11 (Secondary) 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, $π_1(X)=0$, 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. |
| title | Orbit spaces of equivariantly formal torus actions of complexity one |
| topic | Algebraic Topology Combinatorics K-Theory and Homology 57S25, 57N65, 55N91, 55N25, 06A07 (Primary) 55R20, 18G35, 18G10, 55N30, 06A11 (Secondary) |
| url | https://arxiv.org/abs/1912.11696 |