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.
