Stable homotopy equivalent but not homotopy equivalent

Question

Are there known examples of spaces which are stable homotopy equivalent but not homotopy equivalent?

Answer 1

See Suspension of a product - tricky homotopy equivalence and James construction and homotopy splitting: there is a homotopy equivalence $$ \Sigma (X \times Y) \simeq \Sigma X \vee \Sigma Y \vee \Sigma (X \wedge Y) = \Sigma (X \vee Y \vee (X \wedge Y)) $$ Let $X=Y=S^1$. Then $S^1 \times S^1$ is not homotopy equivalent to $S^1 \vee S^1 \vee (S^1 \wedge S^1)$ because they have different ring structures on cohomology, but their suspensions are homotopy equivalent by the above.

Another source of examples ought to be this: take a map $f: S^m \to S^k$ which is not null-homotopic but which becomes null-homotopic upon suspension, like $2 \eta$, where $\eta$ is the Hopf map $S^3 \to S^2$. Then its cofiber, as compared to the cofiber of the zero map (i.e., a wedge of spheres), should be an example.