Pushouts of spaces are not homotopy invariant

Example 2.1.1. Given two maps \(f\colon Z \to X\) and \(g\colon Z \to Y\) of topological spaces, recall that the pushout in \(\Top \) is given by the quotient space \[ X \sqcup _Z Y := (X \sqcup Y) / \sim , \qquadtext { where } f(z) \sim g(z) \text { for all } z \in Z. \] If we let \(D^n\) denote the \(n\)-dimensional disk, and let \(S^{n-1} \hookrightarrow D^n\) denote its boundary, then we have the following two pushout squares in \(\Top \):

These two diagrams are homotopy equivalent, by contractibility of the disk \(D^n\), yet their pushouts \(S^n\) and \(*\) are not homotopy equivalent for \(n \geq 1\).