# homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$ $$F(D^m,k)\simeq F(\mathbb{R}^m,k)$$ or not?

(2). Let $M$ be a manifold. For each $k$, does the $k$-th configuration space on $M\times [0,1]$ homotopy equivalent to the $k$-th configuration space on $M\times (0,1)$ $$F(M\times [0,1],k)\simeq F(M\times(0,1),k)$$ or not?

How to prove these two? Thanks.

• The general statement is that the configuration space of $k$ points on a manifold $M$ with boundary is homotopy equivalent to the configuration space of $k$ points on $\text{int}(M)$. – Qiaochu Yuan Jul 16 '15 at 2:39
• @QiaochuYuan: Thanks so much. Could you give any references/proofs for the general statement? – QSR Jul 16 '15 at 3:22
• See Definition $2.3$, Proposition $2.4$, and Example $2.5$ of these notes. – Michael Albanese Jan 7 '18 at 10:27