cellular homology


If X is a cell space, then let (𝒞*(X),𝔡) be the cell complex where the n-th group 𝒞n(X) is the free abelian groupMathworldPlanetmath on the cells of dimension n, and the boundary mapPlanetmathPlanetmath is as follows: If en is an n-cell, then we can define a map φf:enfn-1, where fn-1 is any cell of dimension n-1 by the following rule: let φ:enskn-1X be the attaching map for en, where skn-1X is the (n-1)-skeleton of X. Then let πf be the natural projectionMathworldPlanetmath

πf:skn-1Xskn-1X/(skn-1X-f)f/f.

Let φf=πfφ. Now, f/f is a (n-1)-sphere, so the map φf has a degree degf which we use to define the boundary operator:

𝔡([en])=dimf=n-1(degφf)[fn-1].

The resulting chain complex is called the cellular chain complex.

Theorem 1

The homologyMathworldPlanetmathPlanetmath of the cellular complex is the same as the singular homology of the space. That is

H*(𝒞,𝔡)=H*(C,).

Cellular homology is tremendously useful for computations because the groups involved are finitely generatedMathworldPlanetmathPlanetmathPlanetmath.

Title cellular homology
Canonical name CellularHomology
Date of creation 2013-03-22 13:14:55
Last modified on 2013-03-22 13:14:55
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 6
Author bwebste (988)
Entry type Definition
Classification msc 55N10