Orbit configuration spaces and the homotopy groups of the pair $(\prod_{1}^{n} M, F_n(M))$ for $M$ either $S^2$ or $RP^2$ - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Israel Journal of Mathematics Année : 2022

Orbit configuration spaces and the homotopy groups of the pair $(\prod_{1}^{n} M, F_n(M))$ for $M$ either $S^2$ or $RP^2$

Résumé

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 $\iota_{n}$, the homotopy fibre $I_{n}$ of $\iota_{n}$ and its homotopy groups, and the induced homomorphisms $(\iota_{n})_{\#k}$ on the $k$th homotopy groups of $F_{n}(M)$ and $\prod_{1}^{n}\, M$ for all $k\geq 1$, where $M$ is the $2$-sphere $\mathbb{S}^{2}$ or the real projective plane $\mathbb{R}P^{2}$. It is well known that the group $\pi_{k}(I_{n})$ is the homotopy group $\pi_{k+1}(\prod_{1}^{n}\, M, F_n(M))$ for all $k\geq 0$. 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 image of the boundary homomorphism $\pi_{k+1}(\prod_{1}^{n}\, M)\longrightarrow \pi_{k}(I_{n})$. From this, if $M=\mathbb{S}^{2}$ and $n\geq 3$ (resp. $M=\mathbb{R}P^{2}$ and $n\geq 2$), we show that $\ker{(\iota_{n})_{\#1}}$ is isomorphic to the quotient of $R_{n-1}$ by the square of 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 $P_{n}(M)$ that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in a previous paper. This paper is a shortened version of "The homotopy fibre of the inclusion $F_{n}(M) \longrightarrow \prod_{1}^{n}\, M$ for $M$ either $\mathbb{S}^{2}$ or the real projective plane $\mathbb{R}P^{2}$ and orbit configuration spaces", ⟨hal-01627001⟩. The main difference between the two is the statement and proof of Theorem 3.
Fichier principal
Vignette du fichier
homotopyfibre_final.pdf (258.33 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03603129 , version 1 (09-03-2022)

Identifiants

  • HAL Id : hal-03603129 , version 1

Citer

Daciberg Lima Gonçalves, John Guaschi. Orbit configuration spaces and the homotopy groups of the pair $(\prod_{1}^{n} M, F_n(M))$ for $M$ either $S^2$ or $RP^2$. Israel Journal of Mathematics, In press. ⟨hal-03603129⟩
70 Consultations
46 Téléchargements

Partager

Gmail Facebook X LinkedIn More