homology of configuration spaces of non-compact manifolds

Question

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?

Answer 1

I believe that, although the results of Bodigheimer-Cohen-Taylor are stated for compact manifolds (potentially with boundary), they hold for noncompact manifolds which are homeomorphic to the interior of a compact manifold. See, for instance, the last sentence of 2.1 of that article.

If $M$ is a compact manifold with boundary, and $N = M \setminus \partial M$ its interior, then the natural inclusion $B(N, n) \to B(M, n)$ is a homotopy equivalence (with inverse induced by an injective map $M \to N$ which ``pushes $M$ in from its boundary;" this is isotopic to the identity). So to compute the cohomology of $B(N, n)$, it suffices to compute that of $B(M, n)$. If mod 2 coefficients are what you're after, Bodigheimer-Cohen-Taylor is precisely what you need.

Regarding your specific question of $S^2 \times \mathbb{R}$, this is homeomorphic to the interior of the compact manifold $S^2 \times [0, 1]$, so BCT applies.