Morse Theory
In the context of handle decompositions of manifolds, attachment refers to the process of connecting lower-dimensional handles to a manifold to construct a more complex topological space. This is done by identifying the boundary of the handle with a specific part of the manifold, allowing for a controlled way to build up the structure of the manifold while preserving its topological properties. Attachment is crucial for understanding how different handles contribute to the overall shape and features of the manifold.
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