纤维化中将Hamilton圈提升到同位素
Lifting Hamiltonian loops to isotopies in fibrations
摘要：
设$G$是李群，$H$是闭子群，$M$是齐性空间$G/H$。$H$的每个表示$Ψ$确定赋予$G$不变连接的$M$上的$G$等变主丛${\mathcal P}$.我们考虑同构群${\rm Diff}(M)$的子群${\mathcal G}$，使得每个向量场$Z\in{\rm Lie}({\mathcal G})$允许提升到${\mathcal P}$上的保持联络向量场.证明了$#\,π_1({\mathcal G})\geq #\,Ψ(Z(G))$ .这个关系适用于半单群$G$的旗簇的哈密尔顿群的子群${\mathcal G}$。设$M_Δ$是由Delzant多胞形$Δ$确定的复曲面流形。我们把$φ_{\bf b}$作为由格向量${\bf b}$定义的$M_Δ$的哈密顿群中的循环。本文给出了回路$φ_{\bf b}$和$φ_{\bf\tilde b}$同伦不等价的一个充分条件，即用$Δ$的质心表示.
英文摘要：
Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $Ψ$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups ${\mathcal G}$ of the diffeomorphism group ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ admits a lift to a preserving connection vector field on ${\mathcal P}$. We prove that $#\,π_1({\mathcal G})\geq #\,Ψ(Z(G))$. This relation is applicable to subgroups ${\mathcal G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_Δ$ be the toric manifold determined by the Delzant polytope $Δ$. We put $φ_{\bf b}$ for the the loop in the Hamiltonian group of $M_Δ$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $Δ$, for the loops $φ_{\bf b}$ and $φ_{\bf\tilde b}$ to be homotopically inequivalent.
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