Abstract
We consider the topological interactions of vortices on general surfaces. If the genus of the surface is greater than zero, the handles can carry magnetic flux. The classical state of the vortices and the handles can be described by a mapping from the fundamental group to the unbroken gauge group. The allowed configurations must satisfy a relation induced by the fundamental group. Upon quantization, the handles can carry "Cheshire charge." The motion of the vortices can be described by the braid group of the surface. How the motion of the vortices affects the state is analyzed in detail.
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