1
$\begingroup$

I am interested in the hyperplane arrangement in $\mathbb{C}^n$ defined by the polynomial $$ (x_i-x_j)(x_i+x_j) $$ for $1<i<j\leq n$.

I vaguely recall that the completion of this arrangement should be a $K(\pi ,1)$-space where $\pi$ should be a braid group of some kind. Where can I look for information on that topic?

Improve this question
$\endgroup$
12
  • 1
    $\begingroup$ This is the arrangement underlying the braid/Coxeter group of type B. That this is a $K(\pi,1)$ is due to Brieskorn in that case maths.ed.ac.uk/~v1ranick/papers/brieskorn8.pdf. $\endgroup$
    – Adrien
    Commented Jan 20, 2021 at 12:34
  • 1
    $\begingroup$ @Adrien And what is $\pi$ in this situation? And isn't it of type D? $\endgroup$
    – Igor Sikora
    Commented Jan 20, 2021 at 12:55
  • $\begingroup$ I also think it's type D. Type B would also include the coordinate hyperplanes, i.e., the polynomial would have each $x_i$ as a factor. $\endgroup$
    – Andreas Blass
    Commented Jan 20, 2021 at 15:02
  • $\begingroup$ Right, type D sorry, this is still handled in Brieskorn paper. $\pi$ is then called the pure braid group of type D. This is the kernel of the natural map from the full braid group of type D to the corresponding Coxeter group, where the former has a presentation obtained from the standard Coxeter presentation of the Coxeter group by removing the torsion relations. There is quite a lot of literature on those, I'm not sure what exactly you'd like to know. $\endgroup$
    – Adrien
    Commented Jan 20, 2021 at 15:07
  • $\begingroup$ @Adrien - the very basics, i.e. the definition of pure braid group of type D (or any other type) and why this is a fundamental group of this arrangement. $\endgroup$
    – Igor Sikora
    Commented Jan 20, 2021 at 15:10

0

Reset to default

You must log in to answer this question.