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

TL;DR

This paper investigates the homotopy fiber of the inclusion of configuration spaces into Cartesian products for spheres and projective planes, revealing their homotopy types and algebraic properties of induced homomorphisms.

Contribution

It characterizes the homotopy fiber and induced homomorphisms for configuration spaces on $\

Findings

Homotopy groups are injective and diagonal for $k extgreater 1$.

Homotopy fiber has the type $K(R_{n-1},1) imes ext{loop space}$.

Kernel of the induced map on $ ext{pi}_1$ relates to the pure braid group quotient.

Abstract

Let $n\geq 1$, and let $\iota\_{n}\colon\thinspace F\_{n}(M) \longrightarrow \prod\_{1}^{n} M$ 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 $(\iota\_{n})\_{#k}$ on the $k$th homotopy groups of $F\_{n}(M)$ and $\prod\_{1}^{n} M$ for $k\geq 1$ in the cases where $M$ is the $2$-sphere $\mathbb{S}^{2}$ or the real projective plane $\mathbb{R}P^{2}$. If $k\geq 2$, 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 $I\_{n}$ 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 $\mathbb{S}^{2}\setminus \{\widetilde{z}\_{0}, -\widetilde{z}\_{0}\}$ with respect to the action of the antipodal map of $\mathbb{S}^{2}$ 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 $\pi\_{k+1}(\prod\_{1}^{n} M)\longrightarrow \pi\_{k}(I\_{n})$. From this, if $M=\mathbb{R}P^{2}$ and $n\geq 2$, we show that $\ker{(\iota\_{n})\_{#1}}$ 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.