Assuming the first two axioms for a homology theory on the CW category, show that the direct limit axiom implies the wedge sum axiom. Show that the converse also holds for countable CW complexes.
Solution
Let \(\widetilde h_*\) satisfy the homotopy and exactness axioms. Recall first that these two axioms already imply the wedge axiom for finite wedge sums.
Assume the direct-limit axiom. Let \[ X=\bigvee_{\alpha}X_\alpha. \] Every finite subcomplex \(K\subset X\) is contained in a finite wedge \[ K_{\alpha_1}\vee\cdots\vee K_{\alpha_r} \] with each \(K_{\alpha_i}\subset X_{\alpha_i}\) finite. Hence, using the finite-wedge result, \[ \widetilde h_n(K) \cong\bigoplus_{i=1}^r\widetilde h_n(K_{\alpha_i}). \] Taking the filtered colimit over finite subcomplexes and using the direct-limit axiom in each summand gives \[ \widetilde h_n\!\left(\bigvee_\alpha X_\alpha\right) \cong\bigoplus_\alpha \widetilde h_n(X_\alpha). \] Thus the wedge axiom follows.
Conversely, assume the wedge axiom and let \(X\) be a countable CW complex. Choose an exhaustion by finite subcomplexes \[ X_1\subset X_2\subset\cdots, \qquad X=\bigcup_iX_i. \] Let \(T\) be the mapping telescope of this sequence. There is a standard cofibration \[ \bigvee_{i\ge1}X_i \xrightarrow{\,1-s\,} \bigvee_{i\ge1}X_i \longrightarrow T, \tag{1} \] where on the \(i\)-th summand the shift \(s\) is the inclusion \(X_i\to X_{i+1}\) followed by inclusion into the \((i+1)\)-st wedge summand. The telescope deformation retracts onto \(X\) up to homotopy, so \(\widetilde h_*(T)\cong\widetilde h_*(X)\).
By the wedge axiom, applying \(\widetilde h_n\) to (1) yields \[ \bigoplus_i\widetilde h_n(X_i) \xrightarrow{\,1-s_*\,} \bigoplus_i\widetilde h_n(X_i) \longrightarrow\widetilde h_n(T) \longrightarrow \bigoplus_i\widetilde h_{n-1}(X_i). \] The map \(1-s_*\) on a direct sum is injective: if a finite-support element lies in its kernel, inspect its first nonzero coordinate and proceed inductively. Therefore exactness gives \[ \widetilde h_n(T) \cong\operatorname{coker}(1-s_*) \cong\varinjlim_i\widetilde h_n(X_i). \] Hence \[ \boxed{\widetilde h_n(X)\cong\varinjlim_i\widetilde h_n(X_i),} \] which is the direct-limit axiom for countable CW complexes.