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I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property.

Let $\Lambda: \mathbb{R}/\mathbb{Z} \rightarrow \mathcal{L}(n)$ and $\Psi :\mathbb{R}/\mathbb{Z} \rightarrow \operatorname{Sp}(2n)$ be two loops, then Maslov index, $\mu$ satisfies: $\mu(\Psi \Lambda)= \mu(\Lambda) +2\mu(\Psi)$

It should be easy but I don't see it, they say that it's implied from the Homotopy property of Maslov index, but I am clueless here.

Any hints?

Any input is more than welcome. Thanks.

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    This is a little more delicate to prove if you take a different construction of the Maslov $in$dex -- for $in$stance, an alternative definition involves counting intersections with the Maslov cycle. The good news is that any index that satisfies the axioms is the Maslov index, so once you have proved that the Maslov index exists, you can use the most convenient one for the calculation you want to do.2012-03-23

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