论文标题
组合拓扑中的立方体和立方连锁链
Cubes and cubical chains and cochains in combinatorial topology
论文作者
论文摘要
本文是作者论文Arxiv的延续：1909.00940 [Math.at]专门研究了Alexander and Sperner的引理，但独立于此。我们从亚历山大（Alexander）和斯派纳（Sperner）到勒布斯格（Lebesgue）的一步开始，从事维度的不变性。与几乎所有人相比，Lebesgue与立方体合作而不是简单。他的方法是由Hurewicz和Lusternik-Schnirelmann开发的，然后被遗忘了。在本文中，这些方法是用立方链和科刀的语言重铸。 此后，我们为Lebesgue和Lusternik-Schnirelmann定理提供了一种新的方法，既是概念性又是基本的。它基于Serre对奇异立方根的产品定义对离散设置的定义。主要结果是新的纯粹组合“立方引理”。这种方法还阐明了Kuhn和Ky Fan的Sperner Lemma的立方版本。特别是，在横向假设下，Ky Fan的引理可以理解为Lebesgue或Kuhn结果的自然增强。 该博览会不假定代数拓扑的任何知识。
The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-Schnirelmann theorems which is both conceptual and elementary. It is based on adaptation of Serre's definition of products of singular cubical cochains to discrete setting. The main results are new purely combinatorial "cubical lemmas". This approach also clarifies the cubical versions of Sperner lemma of Kuhn and Ky Fan. In particular, Ky Fan's lemma can be understood as a natural strengthening of Lebesgue or Kuhn's results under a transversality assumption. The exposition does not assume any knowledge of algebraic topology.
