# Hatcher's proof of the Freudenthal suspension theorem, definition of the suspension map

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?

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 $$h_1 = \Sigma f$$.