Mathematics > Algebraic Topology
[Submitted on 15 Mar 2018 (v1), last revised 2 Apr 2019 (this version, v2)]
Title:The foundations of $(2n,k)$-manifolds
View PDFAbstract:In 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 the 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 an $(2n,k)$-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 rk$G=$rk$H$, contains $(2n,k)$-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$.
Submission history
From: Svjetlana Terzic [view email][v1] Thu, 15 Mar 2018 14:15:18 UTC (28 KB)
[v2] Tue, 2 Apr 2019 09:05:45 UTC (45 KB)
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