# Homotopy invariance of configuration spaces

Let $$\mbox{PConf}_{n}(M)$$ be the configuration space of $$n$$ distinct ordered points in manifold $$M.$$ The symmetric group $$S_{n}$$ acts on $$\mbox{PConf}_{n}(M)$$ by permuting the coordinates. The quotient $$\mbox{Conf}_{n}(M):=\mbox{PConf}_{n}(M)/S_{n}$$ is the unordered configuration space. In the paper ''Configuration spaces are not homotopy invariant'' Longoni and Salvatore proved that configuration spaces (both ordered and unordered) of two homotopically equivalent Lense spaces are not homotopically equivalent. Let $$M$$ and $$N$$ two homotopically equivalent manifolds but not homeomorphic. Are the following statements true?

1. If $$\mbox{PConf}_{n}(M)$$ and $$\mbox{PConf}_{n}(N)$$ are not homotopically equivalent then $$\mbox{Conf}_{n}(M)$$ and $$\mbox{Conf}_{n}(N)$$ are not homotopically equivalent.

2. If $$\mbox{Conf}_{n}(M)$$ and $$\mbox{Conf}_{n}(N)$$ are not homotopically equivalent then $$\mbox{PConf}_{n}(M)$$ and $$\mbox{PConf}_{n}(N)$$ are not homotopically equivalent.

• This is probably more appropriate for math overflow (unless you think that this problem has been studied before and are looking for a reference) Jul 11, 2020 at 23:16

This is not an answer to your questions, however I would recommend to look at the recent paper by Leonid Plakhta https://link.springer.com/article/10.1007/s40879-018-00309-0 where it is proved that the configuration space $$F_n(M)$$ of $$n$$ particles in a compact connected PL manifold $$M$$ with nonempty boundary $$\partial M$$ is homotopy equivalent to the configuration space $$F_n(Int M)$$ of the interior of $$M$$. That paper describes certain types of deformations of a manifold into a submanifod which induce homotopy equivalences of the corresponding configuration spaces. At least this allows to exclude those types of homotopy equivalences of manifolds that do not satisfy assumptions of your questions.