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 Tr. Mat. Inst. Steklova, 2018, Volume 302, Pages 23–40 (Mi tm3930)

Torus actions of complexity 1 and their local properties

Anton A. Ayzenberg

Faculty of Computer Science, National Research University "Higher School of Economics," Kochnovskii proezd 3, Moscow, 125319 Russia

Abstract: We consider an effective action of a compact $(n-1)$-torus on a smooth $2n$-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than $n-1$ has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold $G_{4,2}$, the complete flag manifold $F_3$, and quasitoric manifolds with an induced action of a subtorus of complexity $1$.

 Funding Agency Grant Number National Research University Higher School of Economics 18-01-0030 Ministry of Education and Science of the Russian Federation This work was supported by the HSE Academic Fund Program in 2018–2019 (project no. 18-01-0030) and by the Russian Academic Excellence Project “5-100.”

DOI: https://doi.org/10.1134/S0371968518030020

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 16–32

Bibliographic databases:

UDC: 515.165
MSC: Primary 55R25, 57N65; Secondary 55R40, 55R55, 55R91, 57N40, 57N80, 57S15

Citation: Anton A. Ayzenberg, “Torus actions of complexity 1 and their local properties”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 23–40; Proc. Steklov Inst. Math., 302 (2018), 16–32

Citation in format AMSBIB
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