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The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.

Use the based path space and the path-loop fibration $\Omega S^r\to P{S}^r\to S^r$, where paths begin at a fixed basepoint and the projection evaluates their final endpoint. The total space is contractible: the homotopy $\gamma (t)\mapsto \gamma (st)$, $s\downarrow 0$, contracts every path to the constant path. The fiber is path-connected because $S^r$ is simply connected. Hence $H_0(\Omega S^r)=\mathbb{Z}$ and all positive homology of $P{S}^r$ vanishes.

The exact sequence from part (i), for $i\ge 2$, gives $H_{i-r}(\Omega S^r)\cong H_{i-1}(\Omega S^r)$. Put $d=r-1$. Taking negative-degree homology to be zero, this says $H_j\cong H_{j-d}$ for every $j\ge 1$. Starting with $H_0=\mathbb{Z}$ determines every group:There are no torsion groups hidden in the recurrence, since each step is an actual integral isomorphism. This is the integral homology of the loop space of a sphere.

Choose the unit map $\eta :S^{r-1}\to \Omega S^r$ adjoint to the identity under $\Sigma S^{r-1}\cong S^r$. Loop adjunction gives ${\pi }_j(\Omega S^r)\cong {\pi }_{j+1}(S^r)$, so the loop space is $(r-2)$-connected and ${\pi }_{r-1}(\Omega S^r)\cong \mathbb{Z}$. The adjoint of $\eta$ is the degree-one identity, so $\eta$ sends the generator of ${\pi }_{r-1}(S^{r-1})$ to that generator. The Hurewicz theorem for both $(r-2)$-connected spaces, and its naturality, then show that ${\eta }_{\ast }$ is an isomorphism on $H_{r-1}$. It also is on $H_0$, since both spaces are connected. All other positive degrees below $2r-2$ vanish in both spaces. ThusThis proves the required existence with a specific map, without using the suspension theorem in advance.