奇异交点同调的合理维度概念

Pub Date : 2024-04-04 DOI:10.1007/s40062-024-00343-8
David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré
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引用次数: 0

摘要

M.金(H. King)也从滤波空间的奇异链复数定义了交点同调,并提出了在奇异单纯形中进行选择的关键公式。这个公式需要一个欧几里得单纯形子空间 S 的维度概念,通常是指包含 S 的骨架的最小维度。后来,P. Gajer 使用了另一个维度,基于包含 S 的多面体的维度。在这项工作中,我们证明了西本曼 CS 集的两个相应交点同构是同构的。就 King 的论文而言,这意味着多面体维度是一个 "合理的 "维度。证明使用了 Mayer-Vietoris 论证,需要一个经过调整的细分。多面体维度是一个微妙的问题。一般位置论证是不够的,我们引入了强一般位置。有了强一般位置,一般性质就有了稳定性,我们就可以对每个奇异单纯形进行归纳切割。这种分解是通过伪原点细分实现的,新顶点不是原点,而是原点的近点。
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A reasonable notion of dimension for singular intersection homology

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces S of an Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing S. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing S. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann’s CS sets. In terms of King’s paper, this means that polyhedral dimension is a “reasonable” dimension. The proof uses a Mayer-Vietoris argument which needs an adapted subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.

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