CONNECTEDNESS OF LEVEL SETS FOR NON-DEGENERATE INTEGRABLE SYSTEMS THAT EXTEND COMPLEXITY ONE TORUS ACTIONS

The paper proves Theorem 1.6 rigorously by endowing each nontrivial reduced space with a smooth structure so that the descended function g is Morse and by identifying index‑1 critical points exactly with hyperbolic blocks with connected T‑stabilizer; level‑set connectedness and simple connectivity of the reduced spaces then follow from surface Morse theory and properness of Φ. In contrast, the candidate solution misidentifies the focus–focus case on the reduced surface (it claims a degenerate extremum, whereas the paper shows [p] is a regular point with g◦Ψ^{-1}=y), and it assumes without proof that g descends as a smooth function on the (a priori only topological) reduced surface. These are substantial technical errors even though the final conclusions match the paper’s statements (connected fibers and simply connected reduced spaces). See Theorem 1.6, Proposition 7.1, and Proposition 6.2 for the paper’s precise local models and Morse-theoretic reduction.