A formula of local Maslov index and applications

The paper’s Theorem 1.2 states exactly the Maslov–Morse endpoint formula and the dimension identity the model proves, namely Mas±{λ,μ} = ±dim(λ(0)∩μ(0)∩Y(0)) ∓ dim(λ(1)∩μ(1)∩Y(1)) ± m±(Q(1)) ∓ m±(Q(0)) and dim(λ∩μ) = dim ker Q + dim(A∩B) (). The authors’ proof proceeds via a triangular decomposition and a fixed-vertical reduction (Proposition 5.8), yielding Mas± = ±m±(Q(1)) ∓ m±(Q(0)) and the pointwise dimension identity (), then adds the vertical pair (γ,δ) contribution and telescopes endpoint terms (). Locally, Theorem 1.1 gives that the Maslov jump equals the variation of the Morse indices and dim(λ∩μ)=m0(Q) (). The model supplies a different but compatible route: naturality to a fixed model, graph/linear-relation representation, a short exact sequence yielding dim(λ∩μ) = dim ker Q + dim(A∩B), and a crossing-form/Morse-variation argument on intervals where A,B are fixed. These steps align with identities in the paper (e.g., the identification of Q with ω(x,(λ−μ)x) and the dimension formula (58)) ().