Ordinary homology and cohomology of spheres

Remark 3.3.3. The dimension axiom implies the usual values on spheres. Namely, since \(S^n \simeq \Sigma ^n S^0\) as pointed animae, the suspension isomorphism gives \[ \widetilde H_k(S^n;A) \cong \begin {cases} A & \text {if } k=n, \\ 0 & \text {if } k\neq n, \end {cases} \] and similarly for cohomology.