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# Maps between mapping class groups

Let $X$ be a compact non-orientable surface, maybe with boundary, and let $\tilde X$ be the orienting cover of $X$. If I understand correctly, any smooth automorphism of $X$ lifts naturally to an automorphism of $\tilde X$ (since $\tilde X$ can be viewed as the space of couples (a point $x$ of $X$, an orientation of $T_xX$)). Moreover, composition lifts to composition and isotopic automorphisms lift to isotopic automorphisms. So we get a map $MCG(X)\to MCG(\tilde X)$ where $MCG$ stands for the mapping class group. (In the definition of the mapping class groups we do not require that maps or isotopies should be the identity on the boundary.)

1. Is there a simple way to describe the kernel of the above map?

2. Is it true that the mapping class group of a of non-orientable surface injects into the mapping class group of some orientable surface (not necessarily the orienting cover)?

3. (a slightly unrelated question which nonetheless involves mapping class groups) Is it true that the mapping class group of an orientable surface without boundary injects into the mapping class group of some orientable surface with boundary?