Suspension isomorphism in reduced singular homology

Write $q:X\times [0,1]\to \mathrm{\Sigma }X$ for the quotient. The sets $U=q(X\times [0,2\mathrm{/}3))$ and $V=q(X\times (1\mathrm{/}3,1])$ are open, cover $\mathrm{\Sigma }X$, and contract to their respective apices by moving the height coordinate linearly to $0$ or $1$. Their intersection is $W\cong X\times (1\mathrm{/}3,2\mathrm{/}3)$; projection $p:W\to X$ is a homotopy equivalence, with section at height $1\mathrm{/}2$. Since $X$ has its specified basepoint, these spaces are nonempty.

A point has $H_0(\ast ;G)=G$ and $H_j(\ast ;G)=0$ for $j>0$: its singular complex has one copy of $G$ in each degree, with boundary alternately zero and identity. Thus Contractible nonempty spaces have the homology of a point gives the same groups for $U,V$, with the degree-zero isomorphism induced by augmentation. For $n\ge 1$, ordinary exactness in Mayer–Vietoris sequence in singular homology gives $H_{n+1}(\mathrm{\Sigma }X;G)\cong H_n(W;G)\cong H_n(X;G)$, the last isomorphism by Homotopy equivalences induce isomorphisms on singular homology. These degrees are positive, so the groups are also reduced groups.

For $n=0$, the same exact sequence gives $0\to H_1(\mathrm{\Sigma }X;G)\stackrel{\delta }{\to }{H}_0(W;G)\stackrel{(\epsilon ,-\epsilon )}{\to }G\oplus G$. Its last kernel is exactly ${\tilde{H}}_0(W;G)$. Projection to $X$ preserves augmentation and is an ordinary homology isomorphism, so it identifies this kernel with ${\tilde{H}}_0(X;G)$ under Augmentation at 0-simplices and reduced singular homology.

Every point of $\mathrm{\Sigma }X$ has a height path to an apex, and the height path through $x_0$ joins the two apices. Hence $\mathrm{\Sigma }X$ is path connected: differences of singular points bound paths, so augmentation identifies its $H_0$ with $G$ and its reduced $H_0$ is zero. This proves the shift for $n=-1$; for $n<-1$ both sides are zero by the stated reduced-complex convention. The arguments include the one-point space and $G=0$.

A based map $f:X\to Y$ induces $[x,t]\mapsto [f(x),t]$ and preserves these covers and their projections. At chain level the connecting map takes a small cycle $z=u+v$ to $[\mathrm{\partial }u]$; applying $f_{\mathrm{\#}}$ gives $[\mathrm{\partial }{f}_{\mathrm{\#}}u]=[f_{\mathrm{\#}}\mathrm{\partial }u]$. Thus the connecting maps commute with $f$, and so do their degree-zero kernel restrictions and the projection isomorphisms. This proves naturality in every degree.