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


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:


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


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