Coefficient groups of a generalized cohomology theory

Let $\tilde{h}$ be a reduced generalized cohomology theory on based CW complexes in the sense of Reduced generalized cohomology theory. Its coefficient group in degree $q\in \mathbb{Z}$ is $h^q(\ast ):={\tilde{h}}^q(S^0),$ where $S^0=\{0,1\}$ is based at $1$. Equivalently, if $h$ is the pair theory associated with $\tilde{h}$ by Reduced and unreduced generalized cohomology theories correspond, then ${\tilde{h}}^q(S^0)=h^q(\mathrm{p}\mathrm{t})$: the coefficient group is the value of the unreduced theory on a point.

The suspension isomorphisms of the theory identify the reduced groups of spheres in all degrees. Composing the isomorphisms ${\tilde{h}}^{n-p}(S^0)\stackrel{\sigma }{\to }{\tilde{h}}^{n-p+1}(S^1)\stackrel{\sigma }{\to }\cdots \stackrel{\sigma }{\to }{\tilde{h}}^n(S^p)$ gives, for every $p\ge 0$ and every $n\in \mathbb{Z}$, a canonical isomorphism ${\tilde{h}}^n(S^p)\cong h^{n-p}(\ast )\mathrm{.}$ For $p=0$ this is the identity ${\tilde{h}}^n(S^0)=h^n(\ast )$. These identifications are compatible with the iterated suspension maps used to construct them. An arbitrary based self-map of a sphere need not act as the identity under these identifications; its induced endomorphism is transported to the corresponding endomorphism of the coefficient group. No dimension axiom is imposed, so the groups $h^q(\ast )$ may be nonzero for infinitely many $q$.

Compare Loizides, §2, printed p. 3, for the convention $h^n:=h^n(S^0)=h^n(\mathrm{p}\mathrm{t})$ and Remark 2.2 together with the wedge axiom for the resulting product decompositions.