For a CW pair \((X, A)\) show there is a relative cellular chain complex formed by the groups \(H_i(X^i, X^{i-1} \cup A^i)\), having homology groups isomorphic to \(H_n(X, A)\).
Solution
Let \((X,A)\) be a CW pair, so \(A\) is a subcomplex of \(X\). Define \[ C_n^{CW}(X,A) = H_n(X^n,X^{n-1}\cup A^n), \qquad A^n=A\cap X^n. \]
1. These groups are naturally the cellular chain groups of the quotient CW complex \(X/A\).
The quotient \(X/A\) inherits a CW structure whose non-basepoint cells are exactly the cells of \(X\) not contained in \(A\). Its \(n\)-skeleton is \[ (X/A)^n=X^n/A^n. \] Therefore \[ C_n^{CW}(X/A) =H_n(X^n/A^n,\,X^{n-1}/A^{n-1}). \] By the quotient theorem for relative homology, this is naturally isomorphic to \[ H_n(X^n,X^{n-1}\cup A^n)=C_n^{CW}(X,A). \]
2. Under these identifications, the cellular boundary on \(X/A\) gives boundary maps\[ d_n:C_n^{CW}(X,A)\to C_{n-1}^{CW}(X,A). \]
1. Equivalently, they are the connecting maps obtained from the triples\[ (X^n,\,X^{n-1}\cup A^n,\,X^{n-2}\cup A^{n-1}). \]Since they come from an ordinary cellular chain complex, they satisfy \(d_{n-1}d_n=0\).
3. The homology of this relative cellular chain complex is \(H_n(X,A)\).
By <1>1–<1>2, the relative cellular complex is canonically the cellular chain complex of \(X/A\). Hence its homology is \[ H_n^{CW}(X/A)\cong\widetilde H_n(X/A). \] For a CW pair, collapsing \(A\) gives the standard relative-homology isomorphism \[ \widetilde H_n(X/A)\cong H_n(X,A) \] (with the usual unreduced interpretation in degree \(0\)). Therefore \[ \boxed{H_n(C_*^{CW}(X,A))\cong H_n(X,A).} \]
Thus the groups specified in the problem form the desired relative cellular chain complex.