Making the preceding problem more concrete, construct explicit chain maps \(s: C_n(X) \to C_{n+1}(SX)\) inducing isomorphisms \(\tilde{H}_n(X) \to \tilde{H}_{n+1}(SX)\).
Solution
Let \(i:X\hookrightarrow SX\) be the equatorial inclusion. For a singular simplex \(\sigma:\Delta^n\to X\), let \(C_+\sigma\) and \(C_-\sigma\) be the singular \((n+1)\)-simplices obtained by coning \(i\sigma\) to the upper and lower suspension vertices, respectively, with the cone vertex listed first in the orientation. Define \[ s_n(\sigma)=C_+\sigma-C_-\sigma \] and extend linearly.
1. The cone operators satisfy\[ \partial C_\pm + C_\pm\partial=i_\#. \]
For an oriented simplex \([v_0,\dots,v_n]\), the oriented cone is \([w_\pm,v_0,\dots,v_n]\). Its boundary is \[ [v_0,\dots,v_n]-\sum_{j=0}^n(-1)^j[w_\pm,v_0,\dots,\widehat v_j,\dots,v_n], \] which is exactly \(i_\#\sigma-C_\pm(\partial\sigma)\). This identity extends linearly to all singular chains.
2. Hence\[ \partial s_n=-s_{n-1}\partial. \]Thus \(s\) is a degree-one chain map, equivalently a chain map \(C_*(X)\to C_{*+1}(SX)\) when the shifted complex has differential \(-\partial\).
Subtract the two identities in 1: \[ \partial(C_+-C_-)+(C_+-C_-)\partial=0. \] This is precisely the displayed formula.
3. Therefore \(s\) sends cycles to cycles and boundaries to boundaries, and induces\[ s_*:\widetilde H_n(X)\longrightarrow \widetilde H_{n+1}(SX). \]
If \(\partial z=0\), then \(\partial s(z)=-s(\partial z)=0\). If \(z=\partial c\), then \[ s(z)=s(\partial c)=-\partial s(c), \] so the homology class of \(s(z)\) is zero. The same formulas hold for augmented chains in degree \(0\), hence give the reduced-homology map as well.
4. The induced map \(s_*\) is an isomorphism for every \(n\).
Write \[ SX=C_+X\cup C_-X, \qquad C_+X\cap C_-X=X. \] Both cones are contractible. In the reduced Mayer–Vietoris sequence, the connecting homomorphism \[ \delta:\widetilde H_{n+1}(SX)\longrightarrow\widetilde H_n(X) \] is therefore an isomorphism.
For a cycle \(z\) in \(X\), the chain \(s(z)=C_+z-C_-z\) decomposes as an upper-cone chain plus a lower-cone chain. By the chain-level definition of the Mayer–Vietoris connecting map, \[ \delta([s(z)])=[\partial C_+z]=[z] \] (up to the harmless global sign determined by the convention for \(\delta\)). Thus \(\delta\circ s_*=\pm\operatorname{id}\). Since \(\delta\) is an isomorphism, so is \(s_*\).
Hence the explicit maps \[ \boxed{s_n(\sigma)=C_+\sigma-C_-\sigma} \] induce the suspension isomorphisms \[ \widetilde H_n(X)\cong\widetilde H_{n+1}(SX). \]