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There are a number of specialties in low-dimensional topology that can find in their family tree a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations, and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. However, its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology.