Surfaces of Section for Geodesic Flows of Closed Surfaces

The candidate’s construction matches the paper’s Theorem A: choose 2G waists with the stated intersection pattern (Lemma 4.5), perform Fried surgery on their Birkhoff annuli to obtain a single connected surface of section Σ of genus one with exactly 8G−4 boundary components (Lemma 4.6), and use the no-contractible-without-conjugate-points hypothesis to ensure the complement disk contains no such geodesic so that every orbit segment of uniform length meets Υ and hence Σ (Theorem 4.7 via Lemma 4.2 and Lemma 4.6(iv)). The boundary counting and genus computation agree with the triangulation argument in the paper. Minor gaps in the candidate’s outline (e.g., not explicitly invoking the “convex geodesic polygon” condition, Definition 4.1 of a complete system, or Theorem 3.4 on trapped sets) are standard and covered by the cited lemmas in the paper, which complete the argument as written in the uploaded PDF .