Wrapped Floer homology and the circular restricted three-body problem

The paper proves that, just above the first critical value, HW*(Le,Fix(ρ)) ≅ HW*(Lm,Fix(ρ)) ≅ H*(ΩS^2 × S^1), HW*(Le, Lm) = HW*(Lm, Le) = 0, HW*(Le) ≅ HW*(Lm) ≅ H*(ΩS^2), and HW*(Fix(ρ)) ≅ H*(ΩS^2 × S^1 × S^1) ⊕ H*(ΩS^2 × S^1 × S^1) (Theorem 1.1), by combining Moser regularization/identification in T* S^2 with the invariance of wrapped Floer homology under subcritical handle attachment and the cotangent/conormal path-space computations . The geometry across the first critical value is described as a contact connected sum of two tight RP^3 with a Weinstein 1–handle, consistent with the regularization picture below and above the threshold . The candidate solution reaches exactly the same group-level conclusions by the same structural ingredients: regularization into T* S^2; identification of Le, Lm as cotangent fibers and Fix(ρ) as a conormal to a meridian; and reduction to pre-surgery computations via invariance. The only substantive flaw in the model’s write-up is Step 2, which assumes the 1–handle can be attached away from Fix(ρ); the paper notes the surgery points lie on both pre-surgery components of Fix(ρ), so Fix(ρ) becomes a pair of pants after surgery . For groups involving Fix(ρ), one must use the handle-invariance isomorphism for Lagrangian surgery (Theorem 2.3), rather than a “no-escape” restriction argument . With that correction, the model’s proof aligns with the paper’s and yields the same results. The standard path-space identifications HW*(T*∞S^2, ν* S^1) ≅ H*(ΩS^2 × S^1) and HW*(ν* S^1, ν* S^1) ≅ H*(ΩS^2 × S^1 × S^1) are used on the pre-surgery side, exactly as in the paper’s discussion below the first critical value and in its general wrapped-Floer background .