= Homology comparison proof of Freudenthal suspension
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 $r\geq3$. 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.
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