Handtagslutningar

Handtagslutningar, or handle closures, is a term used in differential topology to describe the closure of a single handle in a handle decomposition of a manifold. In this framework, a smooth n-manifold M is built by attaching handles h^k ≅ D^k × D^{n−k} along their boundary ∂D^k × D^{n−k} to a previously constructed submanifold. The handtagslutning associated with a given handle is the closure in M of the image of the attached k-handle, i.e., the subspace obtained after the attachment, which is homeomorphic to D^k × D^{n−k}.

Construction and properties: Begin with M_0 a manifold (possibly with boundary). Attach a k-handle h^k via an attaching map φ: ∂D^k × D^{n−k} → ∂M_{i−1} or into M_{i−1}. The resulting manifold M_i contains the image of φ on its boundary. The handtagslutning for h^k is the closure of h^k’s image in M_i, and is homeomorphic to D^k × D^{n−k}. Its boundary intersects the rest of the boundary along ∂D^k × ∂D^{n−k}. The process is local, and the global topology of M changes according to the handle’s index k.

Applications: Handtagslutningar are used to analyze how attaching a handle affects topological invariants such as homology and the fundamental group. They are central in Morse theory, surgery theory, and Heegaard splittings, where one tracks the contribution of each handle to the evolving manifold.

Terminology: The term is rooted in Swedish usage; in English literature the same concept is typically called the closure of a handle or simply a handle in a handle decomposition. Different authors may use slight variations in terminology, but the underlying idea is the same.