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Braid groups in complex spaces. (English) Zbl 1309.20030

The fundamental group of the ordered (unordered) configuration space of \(k\) points in \(\mathbb C^n\), denoted by \(\mathcal F_n(\mathbb C^n)\) (\(\mathcal C_n(\mathbb C^n)\)), where \(\mathbb C\) is the complex numbers, is well known. The authors consider a stratification \(\mathcal F_n(\mathbb C^n)=\bigcup_{i=0}^n\mathcal F_k^{i,n}\), where \(\mathcal F_k^{i,n}\) is the ordered configuration space of all \(k\) distinct points \(p_1,\ldots,p_k\) in \(\mathbb C^n\) such that the dimension \(\dim\langle p_1,\ldots,p_k\rangle=i\). A similar strata is defined for \(\mathcal C_n(\mathbb C^n)\) and denoted by \(\mathcal C_k^{i,n}\).
The goal of the paper is to compute the fundamental groups \(\pi_1(\mathcal F_k^{i,n})\) and \(\pi_1(\mathcal C_k^{i,n})\) for all values of \(i,k,n\). Let \(PB_n\) denote the pure Artin braid group of the plane on \(n\) strings and \(B_n\) the full Artin braid group on \(n\) strings.
More precisely they show: Theorem 1.1: The spaces \(\mathcal F_k^{i,n}\) are simply connected except for \(i=1\) or \(i=n=k-1\). In these cases (1) \(\pi_1(\mathcal F_k^{1,1})=PB_k\), (2) \(\pi_1(\mathcal F_k^{1,n})=PB_k/\langle D_k\rangle\) when \(n>1\), (3) \(\pi_1(\mathcal F_{n+1}^{n,n})=\mathbb Z\) for all \(n\geq 1\).
The fundamental group \(\pi_1(\mathcal C_k^{i,n})\) is isomorphic to the symmetric group \(\Sigma_k\) except for \(i=1\) or \(i=n=k-1\). In these cases: (1) \(\pi_1(\mathcal F_k^{1,1})=B_k\), (2) \(\pi_1(\mathcal C_k^{1,n})=B_k/\langle \Delta_k^2\rangle\) when \(n>1\), (3) \(\pi_1(\mathcal C_{n+1}^{n,n})=B_{n+1}/\langle\sigma_1^2=\sigma_2^2=\dots=\sigma_n^2\rangle\) when \(n>1\) .
The element \(D_k\) is the full twist and \(\Delta_k\) is the fundamental Garside braid.
A main ingredient of the proofs is the use of several fibrations which are constructed by geometric means, relating the strata and Grassmannian manifolds. – The paper is well organized and the proofs are clearly written.

MSC:

20F36 Braid groups; Artin groups
14N20 Configurations and arrangements of linear subspaces
57M05 Fundamental group, presentations, free differential calculus
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
51A20 Configuration theorems in linear incidence geometry
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References:

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