Define a candidate for a reduced homology theory on CW complexes by \(\tilde{h}_n(X) = \Pi_i \tilde{H}_i(X) / \oplus_i \tilde{H}_i(X)\). Thus \(\tilde{h}_n(X)\) is independent of \(n\) and is zero if \(X\) is finite-dimensional, but is not identically zero, for example for \(X = \vee_i S^i\). Show that the axioms for a homology theory are satisfied except that the wedge axiom fails.
Solution
Put \[ Q(X)=\prod_{i\in\mathbb Z}\widetilde H_i(X)\big/\bigoplus_{i\in\mathbb Z}\widetilde H_i(X), \qquad \widetilde h_n(X)=Q(X). \]
1. Homotopy invariance holds.
A homotopy equivalence induces isomorphisms on every reduced homology group, hence on their product, their direct sum, and therefore on the quotient \(Q(X)\).
2. Exactness holds, with all connecting maps induced coordinatewise from ordinary homology.
For a cofibration sequence \(A\to X\to X/A\), ordinary reduced homology gives a long exact sequence. Reindexing turns the connecting map into a degree-preserving map between the products defining \(Q\). Products of exact sequences of abelian groups are exact. Direct sums are exact as well. Applying the snake lemma to \[ 0\to\bigoplus_i\widetilde H_i(-)\to\prod_i\widetilde H_i(-)\to Q(-)\to0 \] for the three terms of the cofibration sequence gives exactness for \(Q\).
3. The dimension axiom holds.
For \(S^0\) only one reduced homology group is nonzero. Hence product and direct sum coincide, so \(Q(S^0)=0\) in every degree.
4. The wedge axiom fails.
Let \[ X=\bigvee_{j\ge1}S^j. \] Then \(\widetilde H_i(X)\cong\mathbb Z\) for every \(i\ge1\), so \[ Q(X)\cong\prod_{i\ge1}\mathbb Z\big/\bigoplus_{i\ge1}\mathbb Z\ne0. \] But each individual sphere \(S^j\) is finite-dimensional, hence \(Q(S^j)=0\). Therefore \[ \bigoplus_j\widetilde h_n(S^j)=0 \] while \(\widetilde h_n(\bigvee_jS^j)\ne0\).
Thus the candidate satisfies the reduced homology axioms except the infinite wedge axiom.