A Decomposition Theorem for Topological Semi-small Maps
Abstract
A continuous surjection f: X → Y is an almost covering map if there exists a nowhere dense closed set R ⊂ Y with a finite decomposition R = bigsqcupβ Tβ such that the restriction f|f−1(Y ∖ R) is a covering map and, for each β, the restriction f|f−1(Tβ) 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|f−1(Tβ) and the space Tβ 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 ⊂ Tβ the connecting homomorphisms in the Gysin sequence of the normal bundle of f−1(U) ⊂ X vanish in suitable degrees, the derived direct image of the constant sheaf on the domain admits a decomposition.
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BibTeX
@misc{sharaf2026decomposition,
title = {{A Decomposition Theorem for Topological Semi-small Maps}},
author = {Shahryar Ghaed Sharaf},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30875},
eprint = {2608.30875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30875},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Decomposition Theorem for Topological Semi-small Maps
AU - Shahryar Ghaed Sharaf
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30875
UR - https://arxiv.org/abs/2608.30875
AB - A continuous surjection f: X → Y is an almost covering map if there exists a nowhere dense closed set R ⊂ Y with a finite decomposition R = bigsqcup_β T_β such that the restriction f|_f^−1(Y ∖ R) is a covering map and, for each β, the restriction f|_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|_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 ⊂ T_β the connecting homomorphisms in the Gysin sequence of the normal bundle of f^−1(U) ⊂ X vanish in suitable degrees, the derived direct image of the constant sheaf on the domain admits a decomposition.
ER -