tags: - colorclass/differential geometry ---Cell complexes are fundamental structures in topology and Algebraic Topology, providing a way to construct and analyze spaces in terms of simpler building blocks. These building blocks, or “cells,” are spaces that resemble open disks of various dimensions. By gluing cells together in prescribed ways, one can build up complex spaces from these simpler components, facilitating the study of their topological and algebraic properties.
Definition and Structure
A cell complex is built from cells of various dimensions:
- A 0-cell is a point. - A 1-cell is a line segment (minus the end points), analogous to an open interval in . - A 2-cell is a disk (without the boundary circle), similar to an open disk in . - Similarly, a n-cell is an -dimensional open disk, which can be thought of as an open ball in .
These cells are glued together according to specific rules to form a cell complex. The process involves attaching the boundary of an -cell to a part of the complex that has already been constructed using -cells or lower.
Types of Cell Complexes
- Simplicial Complex: A special type of cell complex where the cells are simplices (points, line segments, triangles, tetrahedra, and their higher-dimensional analogs). Simplicial complexes are widely used due to their combinatorial simplicity.
- CW Complex: Stands for “Closure-finite Weak” topology complexes. CW complexes allow for a more flexible way of building spaces than simplicial complexes. They are defined by a sequence of attaching cells of increasing dimension. This type is particularly useful in algebraic topology for constructing spaces that are difficult or impossible to realize as simplicial complexes.
Construction
The construction of a cell complex typically involves two main steps for each cell being added:
1. Attaching Map: For an -cell to be added to the complex, an attaching map defines how the boundary of the cell (a sphere of dimension ) is attached to the complex’s existing -dimensional skeleton.
2. Quotient Topology: The -cell is then “glued” to the complex using the attaching map, resulting in a new space that includes the -cell as part of the complex. This step is formalized using the quotient topology, where points identified by the attaching map are considered as a single point in the resulting space.
Applications and Importance
- Topology and Geometry: Cell complexes are used to study the properties of spaces, such as their homotopy and Homology Groups, which are central concepts in topology and algebraic topology.
- Computation: Cell complexes, particularly simplicial complexes, are used in computational topology and geometry for tasks like mesh generation and analysis of high-dimensional data.
- Homotopy Type Theory and Higher Inductive Types: In the context of HoTT, cell complexes can be represented using higher inductive types, providing a powerful framework for reasoning about spaces and their properties in a type-theoretical setting.
Cell complexes offer a versatile and powerful way to study and understand the rich structures within mathematical spaces, bridging the discrete and the continuous, and providing tools for both theoretical exploration and practical application in various fields of mathematics and computer science.