No retraction from a disk onto its boundary

Given: The objects and hypotheses in the statement above.

For $n\ge 1$, ${\tilde{H}}_k(S^n;G)$ is $G$ for $k=n$ and $0$ otherwise. For $S^0$, ${\tilde{H}}_0(S^0;G)\cong G$ and all other reduced groups vanish. Thus $H_0(S^n;G)\cong G$ for $n\ge 1$, whereas $H_0(S^0;G)\cong G\oplus G$. (Homology of spheres)

For any abelian group $G$, $H_0(\ast ;G)\cong G$ and $H_n(\ast ;G)=0$ for every integer $n\mathrm{\ne }0$. For every set-indexed family of pairs the canonical map ${⨁}_{\alpha }{H}_n(X_{\alpha },A_{\alpha };G)⟶H_n({⨆}_{\alpha }{X}_{\alpha },{⨆}_{\alpha }{A}_{\alpha };G)$ is an isomorphism. Together with the structural axioms, singular homology is an ordinary theory with coefficient group $G$. (Singular homology satisfies dimension and arbitrary additivity)

If $X$ is a nonempty contractible topological space, then for every $n\ge 0$ and every abelian group $G$, $H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G)\cong H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\ast ;G),$ where $\ast$ denotes a one-point space. (Contractible nonempty spaces have the homology of a point)