拓扑半小映射的分解定理
A Decomposition Theorem for Topological Semi-small Maps
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中文总结 AI 辅助
本文定义拓扑半小映射，研究几乎覆叠的拓扑性质，证明在Gysin序列连接同态消失的条件下，拓扑半小映射的常层导出直像可分解。
中文摘要 AI 辅助
连续满射 $f: X \rightarrow Y$ 若存在无处稠密闭集 $R \subset Y$，其有限分解为 $R = \bigsqcup_{\beta} T_{\beta}$，使得限制映射 $f\vert_{f^{-1}(Y \setminus R)}$ 是覆叠映射，且对每个 $\beta$，限制映射 $f\vert_{f^{-1}(T_{\beta})}$ 是纤维丛，则称其为几乎覆叠映射。若几乎覆叠映射对每个 $\beta$，$f\vert_{f^{-1}(T_{\beta})}$ 的纤维与空间 $T_{\beta}$ 满足代数半小映射对应的相同维数条件，则称其为拓扑半小映射。本文首先研究几乎覆叠的拓扑性质并开发必要工具，随后聚焦拓扑半小映射，证明在如下假设下：对每个 $\beta$ 和 $T_{\beta}$ 的每个平凡化邻域 $U$，$X$ 中 $f^{-1}(U)$ 的法丛的Gysin序列中的连接同态在合适次数下消失，则定义域上的常层的导出直像可分解。
英文摘要
A continuous surjection $f: X \rightarrow Y$ is an almost covering map if there exists a nowhere dense closed set $R \subset Y$ with a finite decomposition $R = \bigsqcup_β T_β$ such that the restriction $f\vert_{f^{-1}(Y \setminus R)}$ is a covering map and, for each $β$, the restriction $f\vert_{f^{-1}(T_β)}$ is a fiber bundle. We refer to an almost covering map as a topological semi-small map if, for each $β$, the fiber of $f\vert_{f^{-1}(T_β)}$ and the space $T_β$ satisfy the same dimension condition as for algebraic semi-small maps. We first study the topological properties of almost coverings and develop the necessary tools. In the next step, we restrict our attention to topological semi-small maps and prove that, under the assumption that for each $β$ and each trivializing neighborhood $U \subset T_β$ the connecting homomorphisms in the Gysin sequence of the normal bundle of $f^{-1}(U) \subset X$ vanish in suitable degrees, the derived direct image of the constant sheaf on the domain admits a decomposition.
