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This article is cited in 15 scientific papers (total in 15 papers)
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 $1$.
Keywords:
torus action, torus representation, Grassmann manifold, complete flag manifold, quasitoric manifold, bundle classification, Hopf bundle, sponge, space of periodic tridiagonal matrices.
Received: March 22, 2018
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, Trudy Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 23–40; Proc. Steklov Inst. Math., 302 (2018), 16–32
Linking options:
https://www.mathnet.ru/eng/tm3930https://doi.org/10.1134/S0371968518030020 https://www.mathnet.ru/eng/tm/v302/p23
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