A handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in opposite directions, pulling the edges of the zips together, and then zipping up.

Handles are to manifolds as cells are to CW-complexes. If  is a manifold together with a -sphere  embedded in its boundary with a trivial tubular neighborhood, we attach a -handle to  by gluing the tubular neighborhood of the -sphere  to the tubular neighborhood of the standard -sphere  in the dim()-dimensional disk. In this way, attaching a -handle is essentially just the process of attaching a fattened-up -disk to  along the -sphere . The embedded disk in this new manifold is called the -handle in the union of  and the handle.

Dyck's theorem states that handles and cross-handles are equivalent in the presence of a cross-cap.

REFERENCES:

Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.