Cellular homology and computations

For a CW complex X, let $X^n$ denote its n-skeleton. With integer coefficients, $C_n^{\mathrm{cell}}(X)=H_n(X^n,X^{n-1};\mathbb Z)$ is free abelian on the oriented n-cells. For an infinite CW complex this is a direct sum: cellular chains still have finite support. The boundary is built by combining the connecting map of a skeletal pair with the next relative map. Exact-sequence compatibility forces consecutive boundaries to compose to zero.

The cellular homology theorem identifies the homology of this complex with singular homology of X. It is not a theorem that the cellular and singular chain groups are identical. The proof uses the special relative homology of cell attachments and the filtration by skeleta. A change of CW structure may alter every displayed matrix while leaving the resulting groups isomorphic. Finite CW structures are particularly convenient because their boundary matrices permit a complete finite integer calculation in each degree.

To find the coefficient of a chosen lower cell in a boundary, take the attaching sphere of an upper cell, map into the preceding skeleton, and collapse everything outside the chosen lower cell to obtain a sphere map. Its degree is the integer coefficient, with orientations fixed. In dimension two, a loop attachment traversing one loop edge several times contributes the signed traversal count. Opposite traversals can cancel in the cellular boundary even when the full attaching word is nontrivial.

For a standard torus cell structure with one vertex, two loop edges, and one face attached by a commutator, the exponent sum of each edge in that word is zero. Both cellular differentials vanish, yielding the familiar free homology groups. This does not make the torus a wedge of two circles and a sphere: the attaching word and later cup product retain additional information. Cellular homology efficiently extracts additive invariants, while a chain matrix alone does not reconstruct the entire homotopy type.

Over the integers, invertible row and column operations must be unimodular to preserve the integral module structure. Smith normal form exposes diagonal divisibility data and therefore finite cyclic summands. Dividing a row by a nonunit integer, as one might do during real Gaussian elimination, changes the integral quotient. A boundary map multiplying by six is injective over the integers but has a six-element cokernel; its real-linear version is an isomorphism and loses that torsion completely.

With coefficients in a ring, tensor the free cellular complex with that ring and compute its homology. Reducing a differential modulo a prime can make it vanish, creating homology in adjacent degrees. It is not generally correct to tensor only the final integer homology groups and stop; the universal coefficient theorem includes a Tor correction. Euler characteristic supplies a useful consistency check for finite complexes through alternating cell counts or field Betti numbers, but it alone cannot detect every torsion group or identify all maps.