# A procedure to determine if an automorphism of a closed 2-manifold extends to an automorphism of a handlebody

In a paper of Casson and Gordon's "A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983), no. 1, 119–137" they give a necessary criterion for a fibred knot to be a ribbon knot.

The criterion is that the monodromy of the bundle has to extend to an automorphism of some handlebody (technicality, the fibre has a circle boundary so one caps-off the surface with a disc to get a closed surface). The handlebody is not known from simply the knot in advance, as it comes from a generalized Dehn lemma that they prove.

My question:

• Q: Given an outer automorphism of the fundamental group of a closed orientable surface, is there an efficient procedure to determine if the automorphism extends to some handlebody that has the surface as boundary?

By the regular 3-manifold Dehn lemma, an automorphism of the surface extends to some handlebody if and only if it preserves the class of curves that bound discs in the handlebody -- which can be phrased entirely in terms of the homomorphism $\pi_1 \Sigma \to \pi_1 H$ where $\Sigma$ is the surface is the boundary of the handlebody $\Sigma = \partial H$. So this is easy to check.

My question has the do with the case where you want to know if such a handlebody exists. Presumably there is a computable procedure for this, but it does not seem immediate to me.

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