Reduced suspension

Definition

Given a topological space with basepoint $(X,x_0)$, the reduced suspension of $X$ is defined in the following equivalent ways:

• It is the smash product of $(X,x_0)$ with $(S^1,p)$ (we could choose $p$ to be any point in $S^1$ (it does not matter since $S^1$ is a homogeneous space).
• It is the quotient of the suspension of $X$ by the identification:

$(x_0,t)\sim (x_0,t^{\prime })$

The reduced suspension of $X$ is denoted by $\mathrm{\Sigma }X$.

Facts

The reduced suspension commutes with joins. In other words:

$\mathrm{\Sigma }(X*Y)=\mathrm{\Sigma }X*\mathrm{\Sigma }Y$