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We give an explicit formula for the signature of handlebody bundles over thecircle in terms of the homological monodromy. This gives a cobounding functionof Meyer's signature cocycle on the mapping class group of a $3$-dimensionalhandlebody, i.e., the handlebody group. As an application, we give atopological interpretation for the generator of the first cohomology group ofthe hyperelliptic handlebody group.

The Meyer function on the handlebody group