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A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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The space of immersions of a loop in a surface

Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions o …
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