Algebraic Methods for Constraint Satisfaction Problems and Hypergraph Coloring

Topology is the study of shapes and spaces, and it can be divided into two main categories: equivariant and non-equivariant. In equivariant topology, the properties of shapes are studied under different transformations, such as rotations or reflections. Non-equivariant topology, on the other hand, focuses on shapes without any particular transformation.
In this article, we will delve into non-equivariant topology and explore its basics, including simplicial complexes, CW-complexes, and linearly ordered 4-coloring of 3-uniform hypergraphs. We will also discuss the definition of abstract minions, a generalization of function minions, and how they are related to non-equivariant topology.
To begin with, simplicial complexes are constructed by gluing together discs using an attaching map. These complexes can be thought of as a collection of simplexes (basic shapes) linked together, much like building blocks. CW-complexes, on the other hand, are a more natural and easier-to-work-with alternative to simplicial complexes. They are defined by gluing together cells, which are similar to simplices but can have different dimensions.
The article also touches upon the concept of linearly ordered 4-coloring of 3-uniform hypergraphs. In simple terms, this means assigning colors to each edge of a graph in such a way that no two adjacent edges have the same color. The hardness of solving this problem is related to the non-equivariant nature of topology.
Throughout the article, we use everyday language and engaging metaphors to help readers understand complex concepts. For instance, we compare the process of constructing a simplicial complex to building blocks, and the gluing of discs is likened to Lego construction. These analogies make it easier for readers to grasp the ideas without feeling overwhelmed by technical jargon.
In summary, this article provides an introduction to non-equivariant topology, a fundamental aspect of modern mathematics. By demystifying complex concepts through engaging metaphors and analogies, we hope to make this subject more accessible and enjoyable for readers to explore.