Dold's Theorem from Viewpoint of Strong Compatibility Graphs

TL;DR

This paper generalizes Dold's theorem by introducing strong compatibility graphs, linking topological properties with graph chromatic numbers, and demonstrates their effectiveness in deriving combinatorial and graph-theoretic results.

Contribution

It introduces strong compatibility graphs to sharpen Dold's theorem, providing a new combinatorial parameter that surpasses dimension in certain cases and enables novel graph constructions.

Findings

Strong compatibility graphs improve bounds in Dold's theorem.

New methods for constructing high chromatic, triangle-free graphs from spheres.

Limitations of topological methods in graph chromatic number determination.

Abstract

Let $G$ be a non-trivial finite group. The well-known Dold's theorem states that: There is no continuous $G$-equivariant map from an $n$-connected simplicial $G$-complex to a free simplicial $G$-complex of dimension at most $n$. In this paper, we give a new generalization of Dold's theorem, by replacing "dimension at most $n$" with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs, called strong compatibility graphs, associated to the target space. Moreover, in a series of examples, we will see that one can hope to infer much more information from this generalization than ordinary Dold's theorem. In particular, we show that this new parameter is significantly better than the dimension of target space "for almost all free $\mathbb{Z}_2$-simplicial complex." In addition, some other applications of strong compatibility graphs will be presented as well. In particular, a new way for constructing triangle-free graphs with high chromatic numbers from an n-sphere $\mathbb{S}^n$, and some new results on the limitations of topological methods for determining the chromatic number of graphs will be given.