Homology of spheres

Put $D={\mathrm{\Delta }}^{n+1}$ and $K=\mathrm{\partial }D$. The integral augmented complex of $D$ has a vertex-cone contraction, as used in A simplex has zero reduced simplicial homology. Tensoring the contraction identity with $G$ preserves it, so the augmented complex with coefficients in $G$ is exact. For $k<n$, its groups and differentials computing reduced homology in degree $k$ agree with those of $K$, hence ${\tilde{H}}_k^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K;G)=0$.

In degree $n$, exactness for $D$ gives $\ker {\mathrm{\partial }}_n=\mathrm{im}{\mathrm{\partial }}_{n+1}$. The map $G=C_{n+1}(D;G)\to C_n(K;G)$ sends $g$ to the alternating sum of its facets with coefficient $g$; it is injective since any one facet has coefficient $g$ or $-g$. There are no $(n+1)$-chains in $K$, so ${\tilde{H}}_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K;G)\cong G$. This uses the augmentation as ${\mathrm{\partial }}_0$ when $n=0$. Above degree $n$ all simplicial groups vanish.

The boundary $K$ of the single simplex is finite, so choose an order on its finite vertex set. The characteristic-simplex comparison of Simplicial and singular homology agree sends each vertex with coefficient $g$ to the corresponding singular point with coefficient $g$; its homology map is independent of this order and commutes with augmentation to $G$. For either theory, ${\tilde{H}}_0=\ker (H_0\to G)$, directly from the kernel-in-degree-zero definition in Augmentation at 0-simplices and reduced singular homology. Thus the ordinary comparison isomorphism restricts to an isomorphism on these kernels; in positive degrees reduced and ordinary homology agree. The calculation therefore transfers to $\mathrm{\mid }K\mathrm{\mid }\cong S^n$, and negative reduced groups are zero by convention.

For $n\ge 1$, the augmentation $H_0(S^n;G)\to G$ is surjective (use any vertex) with zero kernel, hence is an isomorphism. For $n=0$, the simplicial model consists of two vertices and no edges, giving $H_0(S^0;G)\cong G\oplus G$; its augmentation kernel is $\{(g,-g):g\in G\}$. This includes $G=0$.