Suspension operator

 Let $X$ be a topological space. The suspension $SX$ is the quotient space
\[
SX = (X \times [-1, 1]) / {\sim}
\]
where $\sim$ identifies $X \times \{1\}$ to a single point and $X \times \{-1\}$ to another single point. We denote points in $SX$ by $[x, s]$.

Viewing the standard $(p+1)$-simplex $\Delta_{p+1}$ as the topological cone over $\Delta_p$, we can parameterize it by coordinates $(y, t)$ for $y \in \Delta_p$ and $t \in [0, 1]$, where $\Delta_p \times \{1\}$ is collapsed to the apex.

Given a singular $p$-simplex $\sigma \in C_p(X)$, we define two singular $(p+1)$-simplices $S_+\sigma, S_-\sigma \in C_{p+1}(SX)$ corresponding to the upper and lower cones:
\begin{align*}
S_+\sigma(y, t) &= [\sigma(y), t] \quad \text{for } 0 \le t \le 1, \\
S_-\sigma(y, t) &= [\sigma(y), -t] \quad \text{for } 0 \le t \le 1.
\end{align*}
Define the homomorphism $T \colon C_p(X) \to C_{p+1}(SX)$ by taking their formal difference:
\[
T(\sigma) = S_+\sigma - S_-\sigma
\]
$\textbf{Claim:}$ $T$ is a chain map and induces an isomorphism on reduced singular homology groups for all $n \ge 0$:
\[
T_* \colon \tilde{H}_n(X) \xrightarrow{\cong} \tilde{H}_{n+1}(SX)
\]