Cellular chain complex ($C_{\ast }^{\mathrm{cell}}(X)$)

For a CW complex, $C_n^{\mathrm{cell}}(X)=H_n(X^n,X^{n-1})$ is freely generated by the oriented $n$-cells. Connecting maps of skeleton pairs define its differential.

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.

A finite linear combination, with coefficients in $R$, of the oriented $n$-cells of a CW complex.

A cellular chain whose cellular boundary is zero.

A cellular chain that is the boundary of a chain one dimension higher. The chain-complex identity $d_n{d}_{n+1}=0$ makes every cellular boundary a cellular cycle; their quotient is cellular homology.

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.

Collapse two distinct parallel circles of a torus to two separate points. Cutting between the circles gives two annuli, each of which becomes a two-sphere with the two collapsed points as its poles. The resulting CW complex has two vertices, two edges connecting them, and two two-cells with zero cellular boundary. Its homology is as displayed, with all higher groups zero; it is homotopy equivalent to $S^1\vee S^2\vee S^2$.

For a CW complex, the homology of its cellular chain complex naturally equals its singular homology. The relative groups of successive skeleton pairs, free on the cells in the corresponding dimension and zero in other dimensions, yield this identification through their connecting maps.

For a CW complex, applying $\mathrm{Hom}(-,A)$ to its integral cellular chain complex gives a cochain complex. The differential is precomposition with the cellular boundary. For finitely many cells it is represented by the transpose of the boundary matrix, reduced in the coefficient group when appropriate.

The cohomology of the cellular cochain complex naturally agrees with singular cohomology. A CW complex with no odd cells has zero cellular differentials and free integral cohomology on its even cells, although its cup product still requires a separate computation.

If an $n$-dimensional CW complex is extended by exactly one cell in dimension $n+1$, then its cellular cochain group in that degree is the coefficient field $k$. Regardless of how many higher cells are attached, the cohomology is a quotient of a subspace of that one-dimensional group. This gives the displayed bound. Finiteness of the higher cell sets is unnecessary for this particular conclusion.

The cellular boundary ${\mathbb{Z}}^2\to \mathbb{Z}$ is $(m,n)$, with kernel $\mathbb{Z}$ and zero cokernel when the degrees are coprime. The van Kampen theorem gives trivial fundamental group. The Hurewicz theorem supplies a map from the sphere representing a homology generator, and the homological Whitehead theorem makes it a homotopy equivalence. For degrees $2,3$ the space cannot be homeomorphic to a sphere: an embedded circle in a sphere bounds its two complementary regions, so both characteristic boundary maps would have degree of absolute value one.

The coefficient of an $(n-1)$-cell in the cellular boundary of an $n$-cell is the degree of the attaching map after collapsing the complement of that $(n-1)$-cell to obtain a map $S^{n-1}\to S^{n-1}$.