Reduced suspension ($\Sigma X=S^1\wedge X$)

The based suspension collapses both ends and the basepoint cylinder of $X\times I$ to one basepoint. The loop space and reduced suspension are adjoint in based homotopy theory. For a connected based CW complex, the James reduced product models the loop space of its reduced suspension.

For based spaces there is a natural bijection $[\Sigma X,Y]\cong [X,\Omega Y]$. A map on the suspension sends a point $x$ and suspension coordinate $t$ to a loop $t\mapsto f(x,t)$; the collapsed ends and basepoint give based loops. The reverse construction evaluates such loops. The two constructions commute with based homotopies. Applied to spheres it gives ${\pi }_{n+1}Y\cong {\pi }_n\Omega Y$.