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Mat. Sb., 2019, Volume 210, Number 4, Pages 41–86 (Mi msb9106)  

The foundations of $(2n,k)$-manifolds

V. M. Buchstabera, S. Terzićb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Faculty of Science and Mathematics, University of Montenegro, Podgorica, Montenegro

Abstract: The focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of a model space $\mathfrak E$ with an action of the torus $T^k$ is given such that there exists a $T^k$-equivariant homeomorphism $\mathfrak E\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak E/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of a $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where $\operatorname{rk}G=\operatorname{rk}H$, contains $(2n,k)$-manifolds that have nonzero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$.
Bibliography: 23 titles.

Keywords: toric topology, manifolds with torus action, orbit space, moment map, complex Grassmann manifold.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-51-50005-ЯФ_а
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 18-51-50005-ЯФ_а).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9106

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English version:
Sbornik: Mathematics, 2019, 210:4, 508–549

Bibliographic databases:

UDC: 515.164.8+515.164.22+515.165.2
MSC: 57R19, 58E40, 57R91, 52B40
Received: 29.03.2018 and 14.01.2019

Citation: V. M. Buchstaber, S. Terzić, “The foundations of $(2n,k)$-manifolds”, Mat. Sb., 210:4 (2019), 41–86; Sb. Math., 210:4 (2019), 508–549

Citation in format AMSBIB
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