For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then

$$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence

$ (*) $ $$ B(\mathbb{R}^{n+1},2)\simeq B(S^n,2).$$

Inspired by $(*)$, are there any relations between $B(\mathbb{R}^{n+1},k)$ and $B(S^n,k)$ for general $k\geq 3$ or for some particular $k=2^i$?