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An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only consider surface that are orientable. To be more precise,

Question: If $[ϕ]\in Out(F_n)$ is geometric respect to an nonorientable surface, can one always find an orientable surface $S$ such that $[ϕ]$ is also geometric respect to $S$?

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  • $\begingroup$ Even if the answer to the precise question were true, it would not mean that the study can boil down to the orientable case. For instance if we have a geometric pair of outomorphisms (in the sense that they are geometric for a common surface), this would possibly not imply that the latter can be chosen orientable. Still the question is reasonable (though I regret non-orientable surfaces are often considered as bad guys). $\endgroup$
    – YCor
    Commented Jan 28, 2015 at 14:24

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