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?



Null-homotopy of diagonal map
Questions like this are well studied by homotopy theorists, in particular those working in [Lusternik-Schnirelmann category](https://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category) (**warning:** nothing to do with the objects and morphisms kind of category!). In particular, what you are asking for is examples of spaces with weak category equal to 2. The *weak category* of a pointed space $X$ is the least $k$ such that the reduced diagonal $\triangle\_k\colon X\to X\wedge X\wedge\cdots\wedge X = X^{\wedge k}$ is null-homotopic. The weak category was introduced by Berstein and Hilton in [this paper](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-4/issue-3/Category-and-generalized-Hopf-invariants/10.1215/ijm/1255456060.full), as an approximating invariant for LS-category: note that $\mathrm{wcat}(X)\leq\mathrm{cat}(X)$ for any space. (**Warning 2:** Nowadays, the [prevailing convention](https://www.allbookstores.com/Lusternik-Schnirelmann-Category-Mathematical-Surveys/9780821834046) seems to be that these invariants are normalised, so that the weak category of a contractible space is zero.)


Note that the weak category of any (non-contractible) suspension $\Sigma Y$ is $2$. In the paper of Berstein and Hilton linked above they construct spaces $Z$ with weak category $2$ and LS-category $3$ (which therefore cannot be suspensions).


