is configuration space an H-space?

Question

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$.

Is $B(X,n)$ an H-space?

Under what conditions on $X$ will $B(X,n)$ be a H-space?

Note: if $B(X,n)$ is a H-space, then the homology and cohomology are dual Hopf algebras, which is important.

Answer 1

As Olivier Begassat says in the comments, this will typically not be true because the fundamental group will typically fail to be abelian. For example, when $X = \mathbb{R}^2$, the fundamental group is the braid group $B_n$, which is nonabelian for $n \ge 3$.

(In any case there isn't an obvious candidate for an H-space structure.)