Cellular Homology
Homology from a handful of cells: the cellular chain complex, degrees of attaching maps, and the computations that finally make homology practical.
The idea
Cellular homology is a method for computing the homology groups $H_{n}(X)$ of a CW complex, and it is the method that makes those groups computable in practice. Simplicial homology needs a triangulation, which is laborious and not always available. Singular homology is defined for every space, but its chain group in each degree is free on all continuous maps of a simplex into $X$ — an uncountable basis, from which nothing can be computed by hand.
A CW structure is already a finite presentation: a few cells and their attaching maps. Cellular homology converts it directly into a chain complex whose $n$-th group $C_{n}$ is free abelian on the $n$-cells, joined by boundary maps $d$. Where a surface carried uncountably many singular simplices, it now carries one or two cells per dimension.
Theorem (Cellular homology).
Let $X$ be a CW complex, let $C_{n}(X)$ be the free abelian group on its $n$-cells, and let $d_{n} \colon C_{n}(X) \to C_{n-1}(X)$ be the cellular boundary maps. Then for every $n$, $\ker d_{n} \,/\, \operatorname{im} d_{n+1} \;\cong\; H_{n}(X),$ the singular homology of $X$.
In particular the answer does not depend on which cell structure we chose. The computation then rests on the boundary maps, and each entry of one is an integer: the degree of a map of spheres, recording how many times, and with which signs, the boundary of one cell sweeps across another.
Ways to work on it
- Walkthrough. Compute the homology of standard spaces from their cell structures.
- Proof. Why the alternating cell count equals the alternating sum of the Betti numbers, and why that makes the Euler characteristic well defined.
- Practice. Compute boundary maps, homology groups, and Euler characteristics of small complexes.
- Hardest. Compute the full homology of a two-dimensional complex.
Not sure where to start? Take the ten-question placement test.