Let \((X,x_0)\) and \((Y,y_0)\) be based spaces. Identify the wedge sum \(X\vee Y\) with the subspace \(X\times\ts{y_0}\cup\ts{x_0}\times Y\) of \(X\times Y\). The smash product is the quotient space \[ X\wedge Y\coloneqq(X\times Y)/(X\vee Y), \] based at the image of \(X\vee Y\) [@Hat02].
For \(m,n\geq0\) there is a homeomorphism \(S^m\wedge S^n\cong S^{m+n}\), and for a based space \(X\) the reduced suspension satisfies \(\Sigma X\cong S^1\wedge X\) [@Hat02].