Let $M$ be a manifold.

Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.

Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered configuration space.

If $M$ is compact, then the graded vector space structure of $H_*(B(M,n);F)$, where $F=\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$ is a field, is given in the paper *C.-F. BODIGHEIMER, F. COHEN, L. TAYLOR, On the homology of configuration spaces, Topology 1989*.

If $M$ is non-compact, for example, $M=S^2\times \mathbb{R}$, how to understand the graded vector space structure of $H_*(B(M,n);F)$?

Is there any reference or procedure to follow?