Show that homology groups \(H_n^{lf}(X; G)\) can be defined using locally finite chains, which are formal sums \(\sum_\sigma g_\sigma \sigma\) of singular simplices \(\sigma: \Delta^n \to X\) with coefficients \(g_\sigma \in G\), such that each \(x \in X\) has a neighborhood meeting the images of only finitely many \(\sigma\)’s with \(g_\sigma \neq 0\). Develop this version of homology far enough to show that for a finite-dimensional locally compact CW complex \(X\), \(H_n^{lf}(X; G)\) can be computed using infinite cellular chains \(\sum_\alpha g_\alpha e_\alpha^n\).
Solution
A locally finite singular \(n\)-chain is a formal sum \[ c=\sum_\sigma g_\sigma\sigma \] such that every point of \(X\) has a neighborhood meeting the images of only finitely many simplices with nonzero coefficient. The usual singular boundary formula \[ \partial\sigma=\sum_{i=0}^n(-1)^i\sigma|[v_0,\dots,\widehat v_i,\dots,v_n] \] extends termwise to locally finite sums. It remains locally finite, since any neighborhood meeting the image of a face also meets the image of its parent simplex, and each simplex has only finitely many faces. Thus the locally finite chains form a chain complex \[ C_*^{lf}(X;G), \] and we define \[ H_*^{lf}(X;G)=H_*(C_*^{lf}(X;G)). \] Ordinary finite chains embed as a subcomplex. Proper maps induce homomorphisms on locally finite chains, since the inverse image of a compact neighborhood is compact and therefore meets only finitely many simplices in a locally finite family.
Now suppose \(X\) is a finite-dimensional locally compact CW complex. Such a CW complex is locally finite: every point has a neighborhood meeting only finitely many cells. Hence an arbitrary formal cellular chain \[ \sum_\alpha g_\alpha e_\alpha^n \] is locally finite precisely in the geometric sense required above, and the cellular boundary is well-defined termwise. Indeed, near any point only finitely many cells occur, so only finitely many terms can contribute locally.
Filter \(C_*^{lf}(X;G)\) by the skeleta. The relative locally finite homology of successive skeleta is concentrated in the cell dimension: \[ H_k^{lf}(X^n,X^{n-1};G)=0\quad(k\ne n), \] while \[ H_n^{lf}(X^n,X^{n-1};G) \cong\prod_{e_\alpha^n}G, \] the product rather than the direct sum, since infinitely many \(n\)-cells may carry nonzero coefficients. The boundary maps in the exact couples of the skeletal filtration are exactly the usual cellular incidence-number boundary maps, extended to these infinite locally finite sums.
Because \(X\) is finite-dimensional, the skeletal filtration is finite in each total degree, so the standard cellular-homology argument applies without convergence issues. It yields a chain equivalence between \(C_*^{lf}(X;G)\) and the locally finite cellular complex \[ C_n^{lf,cell}(X;G)=\left\{\sum_\alpha g_\alpha e_\alpha^n\right\}, \] with the ordinary cellular boundary formula. Consequently \[ \boxed{H_n^{lf}(X;G) \cong H_n(C_*^{lf,cell}(X;G)).} \] Thus locally finite homology of a finite-dimensional locally compact CW complex is computed by infinite cellular chains.