Торические действия сложности один и их локальные свойства

Труды Математического института им. В.А. Стеклова РАН. 2018. Т. 302. С. 23–40.

Ayzenberg A.

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated xed 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 ltration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n-1 has a specic topology which we axiomatize 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. The introduced notions are elaborated on concrete examples: Grassmann manifold G_{4,2}, complete flag manifold F_3, and quasitoric manifolds with an induced action of a subtorus having complexity one.

Keywords: Grassmann manifold, quasitoric manifold, torus action, действие тора, complete flag variety, torus representation, bundle classification, Hopf bundle, sponge, space of periodic tridiagonal matrices, представление тора, многообразие Грассмана, многообразие полных флагов, квазиторическое многообразие, классификация расслоений, расслоение Хопфа, губка, пространство периодических трехдиагональных матриц

Publication based on the results of: Иследование топологии пространств изоспектральных разреженных матриц (2018)