Singular sphere classes and ordered suspension

Apply [F1] to $A={\mathrm{\Delta }}^d\mathrm{/}\mathrm{\partial }{\mathrm{\Delta }}^d$. A pointed simplicial map corresponds exactly to a continuous pointed map from $\mathrm{\mid }A\mathrm{\mid }=S^d$. Relative homotopies correspond as well: a simplicial homotopy uses ${\mathrm{\Delta }}^1\times A$, whose realization is $[0,1]\times \mathrm{\mid }A\mathrm{\mid }$ by [F2], with the basepoint cylinder collapsed in the pointed version. Thus adjunction induces a bijection of pointed sphere classes. The group identification is the one of [F2]; its multiplication is the usual concatenation of sphere representatives under this correspondence.

The same adjunction argument applies to finite pointed sphere models, including iterated smash powers of $K$. Realization preserves quotients and, by [F2], products with $K$, hence $\mathrm{\mid }K\wedge A\mathrm{\mid }\cong S^1\wedge \mathrm{\mid }A\mathrm{\mid }$. For a simplicial structure map adjoint to $S^1\wedge \mathrm{\mid }\mathrm{Sing}Z\mathrm{\mid }\stackrel{1\wedge {ϵ}_Z}{\to }{S}^1\wedge Z\stackrel{\sigma }{\to }{Z}^{\mathrm{\prime }}$, evaluation of the suspension of $a$ is exactly $\sigma (1\wedge {ϵ}_Z\mathrm{\mid }a\mathrm{\mid })$. The first factor remains the first factor. This proves the claimed compatibility on actual representatives and on their homotopy classes.