1handle

1handle is a term that is usually written as 1-handle in the context of differential topology and Morse theory. It refers to a standard building block used in handle decompositions of smooth manifolds, specifically a index-1 handle. Formally, a 1-handle is the product D^1 × D^{n−1}, where n is the dimension of the ambient manifold. It is attached to a manifold with boundary along its attaching region ∂D^1 × D^{n−1}, which is equivalent to two copies of D^{n−1} embedded in the boundary of the manifold. The core of the handle is the interval D^1 × {0}.

The intuitive idea is that a 1-handle adds a tube-like connection between two parts of the boundary, thereby changing the topology of the manifold. In particular, attaching a 1-handle can connect previously disjoint boundary components or create a tunnel, effectively increasing the “genus” of the boundary in many examples. The process is central to the standard technique of constructing manifolds by successively attaching handles of increasing index, from 0-handles up to k-handles.

A common concrete example occurs in dimension three. Attaching a 1-handle to a 3-ball along two disjoint disks on the boundary yields a solid torus. Repeated 1-handle attachments can produce higher-genus handlebodies, which serve as fundamental pieces in Heegaard decompositions of 3-manifolds. The notion extends to higher dimensions, where 1-handles play a key role in the topology of the resulting manifolds and their boundaries.

Notes: the term is sometimes written with a hyphen as 1-handle; when searching or reading literature, the hyphenated form is more standard. See also 0-handle, 2-handle, and handle decomposition.