Show the isomorphism between cellular and singular homology is natural in the following sense: A map \(f: X \to Y\) that is cellular — satisfying \(f(X^n) \subset Y^n\) for all \(n\) — induces a chain map \(f_*\) between the cellular chain complexes of \(X\) and \(Y\), and the map \(f_*: H_n^{CW}(X) \to H_n^{CW}(Y)\) induced by this chain map corresponds to \(f_*: H_n(X) \to H_n(Y)\) under the isomorphism \(H_n^{CW} \approx H_n\).
Solution
The cellular chain group of a CW complex \(X\) is \[ C_n^{CW}(X)=H_n(X^n,X^{n-1}). \] Let \(f:X\to Y\) be cellular, so \(f(X^n)\subseteq Y^n\) for every \(n\).
1. For each \(n\), \(f\) induces a homomorphism\[ f_n:H_n(X^n,X^{n-1})\longrightarrow H_n(Y^n,Y^{n-1}). \]
Cellularity says precisely that \[ f:(X^n,X^{n-1})\longrightarrow(Y^n,Y^{n-1}) \] is a map of pairs. Relative homology is functorial, so the displayed map is defined.
2. The maps \(f_n\) commute with the cellular boundary operators, hence form a chain map\[ f_\#:C_*^{CW}(X)\to C_*^{CW}(Y). \]
The cellular boundary is the composite \[ H_n(X^n,X^{n-1}) \xrightarrow{\partial} H_{n-1}(X^{n-1}) \longrightarrow H_{n-1}(X^{n-1},X^{n-2}). \] Both arrows are natural with respect to maps of pairs: the first by naturality of the long exact sequence and the second by functoriality of relative homology. Therefore the square \[ \begin{array}{ccc} C_n^{CW}(X)&\xrightarrow{d_n}&C_{n-1}^{CW}(X)\\ \downarrow f_n&&\downarrow f_{n-1}\\ C_n^{CW}(Y)&\xrightarrow{d_n}&C_{n-1}^{CW}(Y) \end{array} \] commutes.
3. Under the canonical isomorphisms\[ H_n^{CW}(X)\cong H_n(X), \qquad H_n^{CW}(Y)\cong H_n(Y), \]the homomorphism induced by the cellular chain map is the ordinary singular-homology map \(f_*\).
The cellular-homology isomorphism is constructed from the long exact sequences of the skeletal pairs and triples \[ (X^n,X^{n-1},X^{n-2}). \] Every map appearing in that construction is a connecting homomorphism, inclusion-induced map, or quotient arising from exactness. All of these are natural. Applying \(f\) gives a morphism between the entire systems of exact sequences for \(X\) and \(Y\), and the comparison maps defining \[ H_n^{CW}(-)\xrightarrow{\cong}H_n(-) \] commute with this morphism. Consequently the diagram \[ \begin{array}{ccc} H_n^{CW}(X)&\xrightarrow{(f_\#)_*}&H_n^{CW}(Y)\\ \downarrow\cong&&\downarrow\cong\\ H_n(X)&\xrightarrow{f_*}&H_n(Y) \end{array} \] commutes.
Thus cellular homology is natural for cellular maps.