TITLE: Null-homotopy of diagonal map
QUESTION [6 upvotes]: For a sphere $S^n$, the diagonal map $\Delta:S^n\to S^n\wedge S^n$ sending $x\mapsto x\wedge x$ is null-homotopic. This is the homotopy group $\pi_n(S^n\wedge S^n)=\pi_n(S^{2n})$ is trivial.
I'm wondering if there are other examples of this phenomenon. That is, is there a non-contractible based space $X$ which is not a sphere, such that the diagonal map $\Delta:X\to X\wedge X$ is null-homotopic?

REPLY [7 votes]: To add to the above: an $r$-connected CW complex $X$ of dimension $\le 3r$ has the homotopy type of a suspension $\Leftrightarrow$ its reduced diagonal map $X \to X \wedge X$ Is null-homotopic. This is the Berstein-Hilton-Ganea theorem.