AI 中文总结
该研究构建四元数环面作用提升的障碍理论，给出全局与局部四元数环面作用可提升的充要条件，阿贝尔情形退化为经典局部环面作用提升理论。
AI 中文摘要
我们研究将全局和局部四元数环面作用提升到主四元数环面丛的问题。设 \\(Q^k=(\operatorname{Sp}(1))^k\cong (S^3)^k\\)，\\(G\\) 是作用在连通、局部有限 CW 复形 \\(X\\) 上的紧李群，\\(Q^k\longrightarrow P\longrightarrow X\\) 是主 \\(Q^k\\)-丛。我们首先构建 Hattori--Yoshida 障碍理论框架的四元数类似物：提升 \\(G\\)-作用的存在性等价于 \\(P\\) 的同构类属于由 Borel 构造 \\(X_G=EG\times_G X\\) 诱导的限制映射的像，特别地，第二陈类存在等变延拓。选定连续伪提升后，其无法定义真正作用的程度由取值于一般非阿贝尔规范群 \\(\mathcal{G}(P)\cong\Gamma(\operatorname{Ad}(P))\\) 的因子集刻画，我们基于该因子集的平凡性得到充要提升判据，并证明存在单个提升时，所有提升模规范共轭的集合由带基点的非阿贝尔 \\(H^1\\)-集分类。随后将该全局理论应用于局部四元数环面作用：拉回至轨道空间的万有覆叠可解开局部 \\(Q^n\\)-作用的扭转，在拉回流形上产生全局定义的作用，该全局作用的初步提升未必与 deck 变换相容，我们定义了取值于规范群的非阿贝尔下降障碍，建立其交叉上闭链恒等式与变换律，并证明原局部作用可提升当且仅当全局提升障碍消失且下降障碍平凡化，在阿贝尔情形下，这些构造退化为经典的局部环面作用提升障碍理论。
英文摘要
We study the problem of lifting global and local quaternionic torus actions to principal quaternionic torus bundles. Let \(Q^k=(\operatorname{Sp}(1))^k\cong (S^3)^n\), let \(G\) be a compact Lie group acting on a connected, locally finite CW complex \(X\), and let \(Q^k\longrightarrow P\longrightarrow X\) be a principal \(Q^k\)-bundle. We first formulate a quaternionic analogue of the obstruction-theoretic framework of Hattori--Yoshida. The existence of a lifted \(G\)-action implies that the isomorphism class of \(P\) lies in the image of the restriction map induced by the Borel construction \(X_G=EG\times_GX\). In particular, the second Chern class admits an equivariant extension. Once a continuous pseudo-lift has been chosen, its failure to define a genuine action is measured by a factor set with values in the generally nonabelian gauge group \(\mathcal G(P)\congΓ(\operatorname{Ad}(P))\). We obtain a necessary and sufficient lifting criterion in terms of the trivializability of this factor set, and show that, once a single lift exists, the set of all lifts modulo gauge conjugacy is classified by a pointed nonabelian \(H^1\)-set. We then apply this global theory to local quaternionic torus actions. Pulling back to the universal covering of the orbit space, untwists a local \(Q^n\)-action and produces a globally defined action on the pulled-back manifold. A preliminary lift of this global action need not be compatible with the deck transformations. We define a gauge-valued nonabelian descent defect, establish its crossed-cocycle identities and transformation law, and prove that the original local action lifts if and only if the global lifting obstruction vanishes and the descent defect is trivializable. In the abelian case, these constructions reduce to the classical obstruction theory for lifts of local torus actions.
Comments38 pages
