A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
Cellular homology computes the homology of a CW complex using a chain complex whose n-th group is free abelian on the n-cells, with boundary operator determined by the degrees of the attaching maps. It produces the same homology as singular homology but with dramatically smaller chain groups — often making hand computation feasible. Cellular homology is the most efficient general-purpose method for computing homology: it combines the flexibility of CW complexes with the computability of finite chain complexes.
Cellular homology is the most practical method for computing the homology of CW complexes. Given a CW complex X with cells en_alpha (n-cells indexed by alpha), the cellular chain group is C_nCW(X) = Znumber of n-cells, the free abelian group with one generator for each n-cell. The cellular boundary operator d_n : C_nCW -> C_{n-1}^{CW} is defined by d_n(en_alpha) = sum_beta d_{alpha,beta} * en-1_beta, where d_{alpha,beta} is the degree of the composite map Sn-1 -> Xn-1 -> Xn-1/Xn-2 = wedge Sn-1 -> Sn-1_beta. This composite takes the attaching map of the n-cell, collapses the (n-2)-skeleton, and projects to the sphere corresponding to the beta-th (n-1)-cell.
The cellular chain complex is derived from the long exact sequence of the pair (Xn, Xn-1). The relative group H_n(Xn, Xn-1) is isomorphic to Znumber of n-cells by excision: Xn/Xn-1 is a wedge of n-spheres (one for each n-cell), and the reduced homology of a wedge of spheres is the direct sum. The connecting homomorphism partial : H_n(Xn, Xn-1) -> H_{n-1}(Xn-1) followed by the quotient map to H_{n-1}(Xn-1, Xn-2) gives the cellular boundary operator. The equality d compose d = 0 follows from the composition of two connecting homomorphisms in the long exact sequence.
The theorem that cellular homology equals singular homology (H_nCW(X) = H_n(X)) is proved by comparing the cellular chain complex to the singular chain complex via the inclusions of skeleta. The proof uses induction: the singular homology of Xn relative to Xn-1 matches the cellular chain group, the boundary maps correspond, and the five lemma ensures the passage from relative to absolute homology preserves the isomorphism.
For practical computation, cellular homology is vastly superior to other methods. The chain groups are small (one generator per cell, rather than one per simplex or one per singular map), and the boundary matrices have entries that are degrees of maps between spheres — computable geometric quantities. For complex projective space CPn (one cell in each even dimension): all boundary maps are zero (since there are no cells in adjacent dimensions), so H_{2k}(CPn) = Z for 0 <= k <= n. For RPn: the cellular chain complex is 0 -> Z -> Z -> ... -> Z, with boundary maps alternating between multiplication by 2 and 0, giving the known homology with Z/2Z torsion in odd dimensions.
Cellular homology also provides the most transparent proof of several key results. The Euler characteristic equals the alternating sum of cell counts (which equals the alternating sum of Betti numbers by the rank-nullity theorem applied to the cellular boundary matrices). The homology of surfaces is easily computed from their standard CW structures (one 0-cell, 2g 1-cells, one 2-cell for genus g). And the effect of attaching a cell is directly visible: attaching an n-cell to X along an attaching map phi : Sn-1 -> X either creates a new n-dimensional homology generator (if phi is null-homotopic) or kills an (n-1)-dimensional homology class (if phi represents a nontrivial class in H_{n-1}).
No topics depend on this one yet.