Cellular cochain complex ($C_{\mathrm{cell}}^q(X;A)=\mathrm{Hom}(C_q^{\mathrm{cell}}(X;\mathbb{Z}),A)$)

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.

For a CW complex $X$ and coefficient abelian group $A$, the degree-$q$ cellular cochain group isChoosing orientations of the $q$-cells identifies it with an arbitrary assignment of one element of $A$ to each cell. Since the cellular chain group is a direct sum over cells, the cochain group is a product ${\prod }_{e^q}A$, which matters when there are infinitely many cells.

The cochain differential is precomposition with the cellular boundary: $(d\varphi )(c)=\varphi (\partial c)$. The identity ${\partial }^2=0$ gives $d^2=0$. In finite cellular bases, its matrix is the transpose of the boundary matrix, interpreted in the coefficient group.

For the circle with one vertex and one edge attached at both ends, $C^0=C^1=A$ and the differential is zero because the two endpoint contributions cancel. This gives $H^0(S^1;A)=H^1(S^1;A)=A$.

For a CW complex $X$ and coefficient abelian group $A$, the cohomology of its cellular cochain complex naturally equals its singular cohomology:The cellular cochain group is $\mathrm{Hom}(C_q^{\mathrm{cell}}(X;\mathbb{Z}),A)$, and its coboundary map is precomposition with the cellular boundary.

If $X$ has only even-dimensional cells, every cellular coboundary map is zero. If, in addition, there are finitely many cells in each dimension, then its integral cohomology groups are free abelian groups, with one generator for each cell in that degree. For infinitely many cells in one degree, the cochain group is a direct product of groups of copies of $\mathbb{Z}$, rather than the direct sum used for cellular chains; the finite-cell freeness conclusion must not be applied without that qualification. Even with all additive groups known, the 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.