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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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It doesn't follow from the homotopy property -- it follows from a direct computation from the definition they give. Here, they define the Maslov index of the loop to be the degree of the composition of their map $\rho$ with $\Lambda$. The question is then to understand how $\rho$ behaves when you compose with a loop of symplectic matrices... but they have already defined the Maslov index for symplectic matrices almost the same way. It's just a question of chasing through these definitions. Let me know and I can walk you through some more details if this doesn't get you unstuck. –  Sam Lisi Mar 23 '12 at 23:13
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This is a little more delicate to prove if you take a different construction of the Maslov index -- for instance, 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. –  Sam Lisi Mar 23 '12 at 23:16
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