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

Let $n\ge 1$, and let $\iota_{n}\colon\thinspace F_{n}(M) \longrightarrow \prod_{1}^{n} M$bethenaturalinclusionofthe$n$thconfigurationspaceof$M$inthe$n$-foldCartesianproductof$M$ with itself. In this paper, we study the map $\iota \mathrm{\_}n$, its homotopy fibre $I\mathrm{\_}n$, and the induced homomorphisms (\iota\_{n})\_{#k} on the $k$th homotopy groups of $F\mathrm{\_}n(M)$ and $\prod \mathrm{\_}{1}^{n}M$ 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 (\iota\_{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 $I\mathrm{\_}n$ has the homotopy type of $K(R\mathrm{\_}n-1,1)\times \mathrm{\Omega }(\prod \mathrm{\_}{1}^{n-1}{\mathbb{S}}^2)$, where $R\mathrm{\_}n-1$ is the $(n-1)$th Artin pure braid group if $M={\mathbb{S}}^2$, and is the fundamental group $G\mathrm{\_}n-1$ of the $(n-1)$th orbit configuration space of the open cylinder $$\mathbb{S}$^{2}\setminus {\widetilde{z}_{0}, -\widetilde{z}_{0}}$withrespecttotheactionoftheantipodalmapof$$\mathbb{S}$^{2}$if$M=$\mathbb{R}$P^{2}$,where$\widetilde{z}_{0}\in $\mathbb{S}$^{2}$. This enables us to describe the long exact sequence in homotopy of the homotopy fibration $I_{n} \longrightarrow F_n(M) \stackrel{\iota_{n}}{\longrightarrow} \prod_{1}^{n} M$ in geometric terms, and notably the boundary homomorphism $\pi_{k+1}(\prod_{1}^{n} M)\longrightarrow \pi_{k}(I_{n})$\mathrm{.}Fromthis,if$M=$\mathbb{R}$P^{2}$and$n\geq 2$,weshowthat$\ker{(\iota_{n})_{#1}}$ is isomorphic to the quotient of $G\mathrm{\_}n-1$ 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 $P\mathrm{\_}n(\mathbb{R}{P}^2)$ that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in a previous paper.