Show that if \(\tilde{h}\) is a reduced homology theory, then \(\tilde{h}_n(\text{point}) = 0\) for all \(n\). Deduce that there are suspension isomorphisms \(\tilde{h}_n(X) \approx \tilde{h}_{n+1}(SX)\) for all \(n\).
Solution
1. One has \(\widetilde h_n(*)=0\) for every \(n\).
The identity map of a point equals the constant map. By the reduced dimension axiom (equivalently, by exactness applied to the trivial pair), the reduced theory has zero coefficient group on a point. Hence \(\widetilde h_n(*)=0\).
2. Let \(CX\) be the cone on \(X\). Then\[ \widetilde h_n(CX)=0 \]for all \(n\).
The cone is contractible, so homotopy invariance and 1 give the result.
3. The pair \((CX,X)\) gives natural isomorphisms\[ \widetilde h_{n+1}(CX,X)\cong\widetilde h_n(X). \]
In the long exact sequence of the pair, the two adjacent groups of \(CX\) vanish by 2, so the connecting map is an isomorphism.
4. Since \(CX/X\cong SX\), excision identifies\[ \widetilde h_{n+1}(CX,X)\cong\widetilde h_{n+1}(SX). \]Combining with 3 yields\[ \boxed{\widetilde h_n(X)\cong\widetilde h_{n+1}(SX)}. \]