A Decomposition Theorem for Topological Semi-small Maps

Abstract

A continuous surjection $f:X\to Y$ is an almost covering map if there exists a nowhere dense closed set $R\subset Y$ with a finite decomposition $R={⨆}_{\beta }{T}_{\beta }$ such that the restriction $f{|}_{f^{-1}(Y\setminus R)}$ is a covering map and, for each $\beta$, the restriction $f{|}_{f^{-1}(T_{\beta })}$ is a fiber bundle. We refer to an almost covering map as a topological semi-small map if, for each $\beta$, the fiber of $f{|}_{f^{-1}(T_{\beta })}$ and the space $T_{\beta }$ 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 $\beta$ and each trivializing neighborhood $U\subset T_{\beta }$ 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.