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I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities".

Let $f\colon \mathbb{C}^n\to \mathbb{C}$ be a (germ of) smooth function and $F\colon \mathbb{C}^n\times \mathbb{C}^k\to \mathbb{C}$ be a deformation of $f$.

Let $\Sigma=\{\lambda\in \mathbb{C}^k~|~ F(-,\lambda)$ has a zero as critical value$\}$.

Proposition 9.1 of aforementioned paper is that $\mathbb{C}^k-\Sigma$ is an Eilenberg-MacLane space and Proposition 9.3 says that its fundamental group is a braid group of some Weyl group associated to $F$.

However, Arnold states these theorems without any proofs. So, I would like to ask where I can find the detailed proofs.

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I think there is a proof in the book "Singularities of differentiable maps, tome II" by Arnold, Varchenko, Gussein-Zade.

There is a Séminaire Bourbaki on this: "Groupes de tresses", by Brieskorn.

In fact I wouldn't be surprised if (part of) the results were due to Deligne, see the famous paper "Les immeubles des groupes de tresses généralisés" ? And maybe also to Brieskorn, with a different proof.

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