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October 2021 Torus actions of complexity one in non-general position
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Osaka J. Math. 58(4): 839-853 (October 2021).


Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly formal (which essentially means that $H^{\rm odd}(X^{2n};\mathbb{Z})=0$)? It happens that homology of the orbit space can be arbitrary in degrees $3$ and higher. For any finite simplicial complex $L$ we construct an equivariantly formal manifold $X^{2n}$ such that $X^{2n}/T^{n-1}$ is homotopy equivalent to $\Sigma^3L$. The constructed manifold $X^{2n}$ is the total space of a projective line bundle over the permutohedral variety hence the action on $X^{2n}$ is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of an action in $j$-general position and prove that, for any simplicial complex $M$, there exists an equivariantly formal action of complexity one in $j$-general position such that its orbit space is homotopy equivalent to $\Sigma^{j+2}M$.

Funding Statement

The article was prepared within the framework of the HSE University Basic Research Program.


The authors thank Mikiya Masuda for sharing his idea to study torus actions in $j$-general position and for his hospitality during our visit to Osaka in 2018. We also thank the anonymous referee for many valuable comments and suggestions which helped to improve the exposition of the paper.


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Anton AYZENBERG. Vladislav CHEREPANOV. "Torus actions of complexity one in non-general position." Osaka J. Math. 58 (4) 839 - 853, October 2021.


Received: 12 July 2019; Revised: 23 June 2020; Published: October 2021
First available in Project Euclid: 11 October 2021

Primary: 14M25 , 52B11 , 57N65 , 57S12 , 57S25
Secondary: 05E45 , 55P10 , 55P40 , 55R10

Rights: Copyright © 2021 Osaka University and Osaka City University, Departments of Mathematics


Vol.58 • No. 4 • October 2021
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