立方体空间纤维化的结构理论
On the structure theory of cubespace fibrations
摘要：
我们研究了立方空间/零空间范畴中的纤维化。我们表明，纤维化的有限程度$f \colon X\rightarrow Y$之间的紧遍历胶合立方空间（特别是零空间）的因素作为一个（可能可数）塔紧交换李群主纤维丛$Y$。如果$f$的结构群是连通的，那么纤维（一致）同构（在强意义上）于一个逆极限的nilmanifold。此外，我们还给出了$f$的纤维同构为子立方体空间的条件。我们引入了任意作用群的极小拓扑动力系统之间关于因子映射的区域邻近等价关系。我们证明了极小远端系统之间的任何因子映射是纤维化，并得出结论，如果这样的映射是有限度的，那么它的因素作为一个（可能可数）塔的主要交换李群紧的扩展，从而实现了Furstenberg的和Bronstein-Ellis结构定理在此设置的改进。
英文摘要：
We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over $Y$. If the structure groups of $f$ are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of $f$ are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg's and the Bronstein-Ellis structure theorems in this setting.
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