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In Corollary 4.24 (Freudenthal Susepension Theroem) of Hatcher - Algebraic Topology, Hatcher says that the suspension map $S:\pi_i(X)\to \pi_{i+1}(SX)$ is same as the map $\pi_i(X)\cong \pi_{i+1}(C_+X,X)\to \pi_{i+1}(SX,C_-X)\cong \pi_{i+1}(SX)$. (Here $C_+X$ and $C_-X$ are respectively the upper and lower cones in $SX$, the isomorphisms are from homotopy long exact sequences (since cones are contractible), and the middle map is induced by inclusion.) Is the composition the definition of $S$? Or is the following correct? Every basepoint-preserving map $f:S^i\to X$ induces a basepoint-preserving map $Sf:SS^{i}=S^{i+1}\to SX$ and $S$ is defined by $[f]\mapsto [Sf]$.

I think the latter seems correct, but I can't see why the latter map is the same as the composition map $\pi_i(X)\cong \pi_{i+1}(C_+X,X)\to \pi_{i+1}(SX,C_-X)\cong \pi_{i+1}(SX)$. Am I missing something?

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I think it's best here to work with reduced suspension. Let \begin{align*}C_+ X &= (X\times [0,1])/(X\times \{1\})\cup (\{*\}\times [0,1]),\\ C_- X&= (X\times [-1,0])/(X\times \{-1\})\cup(\{*\}\times[-1,0])\\ \Sigma X&= (X\times [-1,1])/(X\times\{\pm 1\})\cup (\{*\}\times [-1,1]).\end{align*} Think of each of these as pointed, with the selected point being the image of the subspace we crush under the quotient map. Let $q\colon X\times [-1,1]\to \Sigma X$ denote the quotient map. Fix a representative in $\pi_i(X)$, which we think of as a map $f\colon (I^i,\partial I^i)\to (X,*)$. We then define $$\Sigma f\colon (I^{i+1},\partial I^{i+1})\to (\Sigma X,*)$$ by $$ \Sigma f(x,s) =q(f(x),2s-1),\qquad x\in I^i,\,s\in I. $$ (Here the unsightly $2s-1$ term is to map $I$ homeomorphically onto $[-1,1]$). This descends to a map between homotopy groups $\Sigma\colon \pi_i(X)\to \pi_{i+1}(\Sigma X).$ We want to see this is the same as the composition $$\pi_i(X)\cong \pi_{i+1}(C_+X,X)\to \pi_{i+1}(\Sigma X,C_{-}X)\cong \pi_{i+1}(\Sigma X)$$ where the two isomorphisms come from the LES of pairs, and the middle map is induced by inclusion. Again fix $f\colon (I^i,\partial I^i)\to (X,*)$. We then define $$C_+f\colon (I^{i+1},\partial I^{i+1}, J^{i+1})\to (C_+ X, X,*)$$ by $$C_+f(x,s) = q(f(x),s),\qquad x\in I^i,\,s\in I.$$ Then under the map $\partial\colon \pi_{i+1}(C_+X,X)\to \pi_i(X)$ we see that $[C_+f]$ is sent to $[f]$. Thus it remains to show $C_+f$ and $\Sigma f$ are homotopic through maps $(I^{i+1},\partial I^{i+1}, J^{i+1})\to (\Sigma X, C_-X, *)$. Define $h_t\colon I^{i+1}\to \Sigma X$ by $$h_t(x,s) =(f(x), (1+t)s-t),\qquad x\in I^i,\,s,t\in I.$$ Then $h_t$ is the desired homotopy, with $h_0 = C_+f$ and $h_1 = \Sigma f$.

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