Good evening!
In the construction of Floer homology, one shows a formula that connects the Maslov index/ Conley-ZehnderConley–Zehnder index $\mu$ with the dimension of the moduli spaces of connecting gradient flow lines: $$\operatorname{dim} \mathcal{M}(x,y) = \mu(x)-\mu(y)$$$$\dim \mathcal{M}(x,y) = \mu(x)-\mu(y).$$ In the final step for this formula, a closed symplectic path consisting of four separate curves is chosen, so that the index along each of them is constant -— and then, one uses that the path is contractible, in order to show that the sum of these indices must equal zero, so one can solve for the one that is wanted. See e.g. https://people.math.ethz.ch/~salamon/PREPRINTS/floer.pdfSalamon - Lectures on Floer homology, "proof of theorem 2.2", p. 23. According to Salamon, it is obvious that this path is contractible in $SP(n)^*$, i.e. it has Maslov index $0$. Other sources also just state this fact, e.g. https://arxiv.org/pdf/1802.06435.pdfWeber - Topological Methods in the Quest for Periodic Orbits, p. 87.
But frankly, I don't really see why it should have this property -— so I would be really glad if someone could give me a short answer or idea.
I know that this is a very specific question, but I would still be very happy if anyone could pitch in their ideas.
