Contiguity distance between simplicial maps: Stability, product–wedge principles, and relative variants

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2026-03-15 Epub Date: 2026-01-29 DOI:10.1016/j.topol.2026.109746

Đặng Võ Phúc

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引用次数: 0

Abstract

We develop several structural results about the contiguity distance

SD (φ, ψ)

between simplicial maps

φ, ψ : K \to L

, complementing and extending the framework introduced in Borat et al. (2023) [3]. Our main contributions are: (i) a stability theorem asserting that, for finite complexes, after sufficiently many barycentric subdivisions the discrete contiguity distance agrees with the homotopic distance between the geometric realizations; (ii) multiplicative “triangle-type” bounds for SD obtained from product covers; (iii) a product inequality for the categorical product of simplicial complexes; (iv) a max formula for wedge sums; (v) homological lower bounds transferred from the continuous homotopic distance; and (vi) a relative (excision-like) principle useful when maps coincide on a subcomplex. We also provide toy examples on small complexes (cycles, wedges, and triangles) that make the bounds and equalities explicit.

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简单图之间的邻近距离：稳定性、产品楔原则和相对变体

我们开发了几个关于简单映射φ，ψ:K→L之间的连续距离SD（φ,ψ）的结构结果，补充和扩展了Borat et al.(2023)[3]中引入的框架。我们的主要贡献是：(i)一个稳定性定理，该定理表明，对于有限复合体，经过足够多的质心细分后，离散连续距离与几何实现之间的同伦距离一致；（ii）从产品盖得到的SD的乘法“三角形”界；(3)单纯复合体的范畴积的积不等式；（iv）楔形和的Max公式；(v)连续同伦距离传递的同伦下界；(6)当映射在子复合体上重合时，一个相对的（类似于切除的）原则是有用的。我们还提供了一些关于小复合体（圆、楔形和三角形）的玩具样例，这些例子使边界和等式显式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.