While working with joins and smash products (wedge sums) of topological spaces, I noticed that all examples I know (and find in textbooks) are either involving discrete spaces or spheres, e.g. that $S^m \wedge S^n \simeq S^{m+n}$ and $S^m * S^n \simeq S^{m+n+1}$ or more generally that $X \wedge S^k \simeq \Sigma^k X$, the $k$-fold suspension of $X$.
Does anyone know explicit examples of spaces $X$ and $Y$ which are neither discrete nor contractible nor spheres and such that $X \wedge Y$ or $X *Y$ has the homotopy type of some explicitly (i.e. not just as a quotient space of $X\times Y \times [0,1]$) given space?
Nice example are homology spheres. These are $n$-manifolds having the same homology groups as $S^n$. See https://en.wikipedia.org/wiki/Homology_sphere.
A well-known theorem says that the double suspension of a homology $n$-sphere is homeomorphic to $S^{n+2}$. See https://en.wikipedia.org/wiki/Double_suspension_theorem.
Further examples are obtained via wild arcs $A \subset S^n$. The quotient space $S^n/A$ is not a manifold, but we have $S(S^n/A) \approx S^{n+1}$. See Chapter 2 of
Daverman, Robert J., and Gerard Venema. Embeddings in manifolds. Vol. 106. American Mathematical Soc., 2009.