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(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.

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    $\begingroup$ 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 $\text{int}(M)$. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 16, 2015 at 2:39
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    $\begingroup$ @QiaochuYuan: Thanks so much. Could you give any references/proofs for the general statement? $\endgroup$
    – QSR
    Commented Jul 16, 2015 at 3:22
  • $\begingroup$ See Definition $2.3$, Proposition $2.4$, and Example $2.5$ of these notes. $\endgroup$
    – Michael Albanese
    Commented Jan 7, 2018 at 10:27

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