The homotopy fibre of the inclusion $F\_n(M) \lhook\joinrel\longrightarrow \prod\_{1}^{n} M$ for $M$ either ${\mathbb{S}}^2$ or$\mathbb{R}{P}^2$ and orbit configuration spaces

Abstract

Let $n\ge 1$, and let $\iota \_n:F\_n(M)⟶\prod \_1^{n}M$ be the natural inclusion of the $n$th configuration space of $M$ in the $n$-fold Cartesian product of $M$ with itself. In this paper, we study the map $\iota \_n$, its homotopy fibre $I\_n$, and the induced homomorphisms $(ι\_{n})\_{#k}$ on the $k$th homotopy groups of $F\_n(M)$ and $\prod \_1^{n}M$ for $k\ge 1$ in the cases where $M$ is the $2$-sphere ${\mathbb{S}}^2$ or the real projective plane $\mathbb{R}{P}^2$. If $k\ge 2$, we show that the homomorphism $(ι\_{n})\_{#k}$ is injective and diagonal, with the exception of the case $n=k=2$ and $M={\mathbb{S}}^2$, where it is anti-diagonal. We then show that $I\_n$ has the homotopy type of $K(R\_n-1,1)\times \Omega (\prod \_1^{n-1}{\mathbb{S}}^2)$, where $R\_n-1$ is the $(n-1)$th Artin pure braid group if $M={\mathbb{S}}^2$, and is the fundamental group $G\_n-1$ of the $(n-1)$th orbit configuration space of the open cylinder ${\mathbb{S}}^2\setminus \{\tilde{z}\_0,-\tilde{z}\_0\}$ with respect to the action of the antipodal map of ${\mathbb{S}}^2$ if $M=\mathbb{R}{P}^2$, where $\tilde{z}\_0\in {\mathbb{S}}^2$. This enables us to describe the long exact sequence in homotopy of the homotopy fibration $I\_n⟶F\_n(M)\stackrel{\iota \_n}{⟶}\prod \_1^{n}M$ in geometric terms, and notably the boundary homomorphism $\pi \_k+1(\prod \_1^{n}M)⟶\pi \_k(I\_n)$. From this, if $M=\mathbb{R}{P}^2$ and $n\ge 2$, we show that $\ker{(ι\_{n})\_{#1}}$ is isomorphic to the quotient of $G\_n-1$ by its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order $2$ generated by the centre of $P\_n(\mathbb{R}{P}^2)$ that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in a previous paper.