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:

$(x_{0},t)\sim (x_{0},t')$

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

Facts

The reduced suspension commutes with joins. In other words:

$\Sigma (X*Y)=\Sigma X*\Sigma Y$