Geodesic
Finite complex reflection arrangements are $K(\pi,1)$
David Bessis1
1 École Normale Supérieure, Paris, France
Annals of mathematics, Tome 181 (2015) no. 3, pp. 809-904.

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Let $V$ be a finite dimensional complex vector space and $W\subseteq \mathrm{GL}(V)$ be a finite complex reflection group. Let $V^{\rm reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\rm reg}$ is a $K(\pi,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about $\pi_1(W\backslash V^{\rm reg})$, the braid group of $W$. This includes a description of periodic elements in terms of a braid analog of Springer’s theory of regular elements.
DOI : 10.4007/annals.2015.181.3.1

David Bessis 1

1 École Normale Supérieure, Paris, France 
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David Bessis. Finite complex reflection arrangements are $K(\pi,1)$. Annals of mathematics, Tome 181 (2015) no. 3, pp. 809-904. doi : 10.4007/annals.2015.181.3.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.3.1/

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