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I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:

enter image description here (The proof is not finished yet but I am very confused by now.)

I couldn't understand the proof at all, will really appreciate if anyone could shed some lights on the proof. It seems to me that the authors did not prove that the Artin's representation is given by the composite $E\circ I$, and I have no idea why $P_{n+1}$ is isomorphic to the pullback. Here are some information that might be useful:

  • Let $G$ be a group and let $\mathrm{Aut}(G)$ be the automorphism group of $G$. The holomorph of $\mathrm{Hol}(G)$, is defined as follows:

  • As a set, $\mathrm{Hol}(G)=\mathrm{Aut}(G)\times G$;

  • For each $x,y\in G$ and $f,g\in\mathrm{Aut}(G)$, the multiplication on $\mathrm{Hol}(G)$ is defined by $$(f,x)\cdot(g,y)=(fg, g^{-1}(x)y)\text{.}$$
  • The map $A:B_n\to \mathrm{Aut}(F_n)$ is the Artin's representation of braid groups, where $B_n$ is the braid group, $P_n$ is the pure braid group, $F_n$ is the free group of rank $n$.

  • The corollary is after the following lemma: enter image description here

  • Theorem 2.3: enter image description here

Here I am not asking for an alternative proof of the result, but how to understand the proof presented.

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