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A real 3- or 4-manifold has by definition an orientation preserving smoothinvolution acting on it. We consider Lefschetz fibrations of 4-dimensionalmanifolds-with-boundary and open book decompositions on their boundary in theexistence of a real structure. We prove that there is a real open book whichcannot be filled by a real Lefschetz fibration, although it is filled bynon-real Lefschetz fibrations.

A real open book not fillable by a real Lefschetz fibration