射影平面上的辫群 $B_{n,m}(\mathbb{R}P^2)$ 与广义 Fadell-Neuwirth 短正合列的分裂问题
几何拓扑
2022-08-09 v2 群论
摘要
设 n , m ∈ N n,m\in \mathbb{N} n , m ∈ N B n , m ( R P 2 ) B_{n,m}(\mathbb{R}P^2) B n , m ( R P 2 ) ( n + m ) (n + m) ( n + m ) S n + m S_{n+m} S n + m S n × S m S_n\times S_m S n × S m 1 → B m ( R P 2 ∖ { x 1 , … , x n } ) → B n , m ( R P 2 ) → q ˉ B n ( R P 2 ) → 1 , 1\rightarrow B_m(\mathbb{R}P^2 \setminus \{x_1,\dots,x_n\})\rightarrow B_{n,m}(\mathbb{R}P^2)\xrightarrow{\bar{q}} B_n(\mathbb{R}P^2)\rightarrow 1, 1 → B m ( R P 2 ∖ { x 1 , … , x n }) → B n , m ( R P 2 ) q ˉ B n ( R P 2 ) → 1 , q ˉ \bar{q} q ˉ m m m q : F n + m ( R P 2 ) / S n × S m → F n ( R P 2 ) / S n q:F_{n+m}(\mathbb{R}P^2)/S_n\times S_m\to F_{n}(\mathbb{R}P^2)/S_n q : F n + m ( R P 2 ) / S n × S m → F n ( R P 2 ) / S n F n ( R P 2 ) F_n(\mathbb{R}P^2) F n ( R P 2 ) R P 2 \mathbb{R}P^2 R P 2 n n n n = 1 n=1 n = 1 q ˉ \bar{q} q ˉ q q q n = 2 n=2 n = 2 q ˉ \bar{q} q ˉ q q q n ≥ 3 n\geq 3 n ≥ 3 q ˉ \bar{q} q ˉ q q q m ≡ 0 , ( n − 1 ) 2 mod n ( n − 1 ) m\equiv 0, (n-1)^2\ \textrm{mod}\ n(n-1) m ≡ 0 , ( n − 1 ) 2 mod n ( n − 1 ) q ˉ \bar{q} q ˉ q q q m = k n ( 2 n − 1 ) ( 2 n − 2 ) m=kn(2n-1)(2n-2) m = k n ( 2 n − 1 ) ( 2 n − 2 ) k ≥ 1 k\geq1 k ≥ 1 m = 2 n ( n − 1 ) m=2n(n-1) m = 2 n ( n − 1 ) m ≥ 3 m\geq3 m ≥ 3 B m ( R P 2 ∖ { x 1 , … , x n } ) B_m(\mathbb{R}P^2\setminus\{x_1,\dots,x_n\}) B m ( R P 2 ∖ { x 1 , … , x n }) m ≥ 5 m\geq 5 m ≥ 5
引用
@article{arxiv.2111.07838,
title = {The braid groups $B_{n,m}(\mathbb{R}P^2)$ and the splitting problem of the generalised Fadell-Neuwirth short exact sequence},
author = {Stavroula Makri},
journal= {arXiv preprint arXiv:2111.07838},
year = {2022}
}
备注
34 pages, 15 figures