Assume topological spaces to be normal and paracompact.
Following the article: "The genus of a fiber space" by A. Schwarz, we call the sectional category (or Schwarz genus) of a locally trivial fiber bundle the smallest cardinal of its trivializing cover.
In the same article Schwarz pointed out that the sectional category of the principal bundle coming from Milnor construction: $E_n=G\ast\ldots\ast G$, i.e. a join of $n+1$ copies of a topological group $G$, is $\leq n$.
Question 1: In what situations we have that the sectional category of the $n$-th step of the Milnor join construction is equal exactly $n$?
Question 2: Is there any other topological invariant equal $n$ for the $n$-th step of the Milnor join construction for a group $G$?