ON DETERMINING THE HOMOLOGICAL CONLEY INDEX OF POINCARÉ MAPS IN AUTONOMOUS SYSTEMS

The paper’s Theorem 2.1 states that if one has an index pair (N,L) for φ_h and a basis of cycles u_j in (N_0,L_0) with h‑movable contiguity relations u_j×0 − (∑_i a_{ij}u_i)×1 = ∂c_j + d_j in the lifted covering, then the homological Conley index CH(S_0, Π) equals the conjugacy class of the Leray reduction of the matrix [a_{ij}] (see the abstract and Theorem 2.1) . The paper builds auxiliary index pairs for the translation system and Poincaré map and then constructs homomorphisms γ, δ making the requisite diagrams commute, applying a Leray reduction functoriality proposition to conclude RH(Φ′_1) ≅ RA and hence CH(S_0, Π) equals the conjugacy class of RA . The candidate solution mirrors this approach: it passes from φ_h to Π via the translation system Φ, uses h‑movable contiguous cycles to define A on H(N_0,L_0), constructs intertwiners between H(Π) and A, and invokes functoriality of Leray reduction to identify the Conley index with RA. Minor gaps in the candidate (e.g., not explicitly introducing the quotient-by-period space Y and the “shift” parameter r = k+m used in the paper’s Proposition 1.1 application) do not change the substance; they reflect omitted technicalities rather than a different idea. The definition of h‑movable contiguous cycles and its role are consistent with Section 1.7 of the paper . Overall, both are correct and essentially the same proof strategy.