Birkhoff sections and orbit spaces of Anosov flows

An Anosov flow in dimension 3 can be studied using transverse surfaces called Birkhoff sections. We consider the sign of the boundary components of a Birkhoff section to deduce topological properties of the flow.
Theorem. [Masayuki Asaoka [1], T.M [2], Christian Bonatti 2021'] An Anosov flow on a closed oriented 3-manifold admits a positive Birkhoff section if and only if it is $\mathbb{R}$-covered and positively skewed.
I will discuss one proof of the theorem together with some relations with Fried-Goodman surgeries and Reeb flows. Notice that this theorem admits two other independent proofs found by Masayuki Asaoka [1] and Christian Bonatti.
[1] Masayuki Asaoka, Goodman-Fried surgery, Birkhoff section, and R-covered Anosov flows, arXiv 2108.08215, 2021
[2] Théo Marty, Flots d'Anosov et sections de Birkhoff, thesis manuscript, 2021