Inversion map of a Lie groupoid is a diffeomorphism

Question

The inversion map $Inv:\mathcal G\to \mathcal G$ of a Lie groupoid $\mathcal G$ is given by $Inv(g)=g^{-1}$. And I want to show this inversion map is a diffeomorphism. Any suggestions will be appreciated.

Answer 1

• The graph of $Inv$ equals the pre-image of the submanifold of units $M$ under the multiplication map $Mult:\mathcal{G}^{(2)} \to \mathcal{G}$
• $Mult$ is a smooth submersion, so the preimage of $M$ is a submanifold of $\mathcal{G}^{(2)}$ which is a submanifold of $\mathcal{G}\times \mathcal{G}$.
• If the graph of a map is an embedded submanifold then the map must be...