A CRITERION TO DETECT A NONTRIVIAL HOMOLOGY OF AN INVARIANT SET OF A FLOW IN R3

The paper’s Theorem 1 states that for an isolating block N with connected boundary, if H^1(K)=0 (Čech, Z2) then N is a handlebody and admits a complete cut system whose boundaries meet the tangency curves exactly twice; the proof constructs flow-generated disks from arcs in N^i avoiding n+, shows their dual classes generate ker(H^1(N)→H^1(K)), cuts along finitely many to obtain N′ with H^1(N′)=0, and concludes ∂N′≅S^2 ⇒ N′ is a 3–ball via polyhedral Schönflies . The candidate solution mirrors this structure closely. Its main gaps are technical: it omits the needed hypothesis H2(N,K)=0 (which the paper derives from ∂N connected and H^1(K)=0) and does not mention excluding n+ when choosing arcs. It also misphrases the arcs as “properly embedded in Int N^i, disjoint from τ,” which conflicts with proper embedding; the paper correctly uses arcs properly embedded in N^i with endpoints on τ and disjoint from n+ (continuity of exit time to ensures the disk construction) . These are fixable clarifications rather than substantive differences. Overall, both are correct, with the model following the paper’s proof in substance.