# homology of configuration spaces of non-compact manifolds

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?

• Over the rationals, you can take a look at Ben Knudsen's recent work: arxiv.org/abs/1405.6696. Alternatively, in a range depending on the number of particles you can use homological stability and scanning. This works particularly well over the rationals, where rational homotopy theory is available. Feb 1, 2015 at 3:41
• How about Z/2Z? I mostly want to know Z/2Z. Feb 1, 2015 at 5:58

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.