The connected sum operation allows us to create a (possibly) new surface out of two surfaces we already know. To make a connected sum, we first cut a hole into each of our surfaces, forming a single closed edge on each one. Stretching these edges towards each other, we then line them up and then join the surfaces along the edges.
Connected Sum of two Spheres
The connected sum of two spheres actually yields a single sphere at the end.
As a large portion of this site is dedicated to transforming surfaces from one representation to another, we need first to understand what we are "allowed" to do. In 1985, Tatsuo Homma and Teruo Nagase classified six of these deformations, and in 1998, Dennis Roseman produced seven similar deformations (5 of which were identical to the Homma-Nagase ones). The following movies demonstrate Roseman's seven moves.
Type I Move
This first move is simply pushing one surface through a single other surface.
Type II Move
In this move we push a single surface through a "double curve", the intersection of two other surfaces.
Type III Move
This move pushes a surface through a "triple point", where three surfaces intersect.
Type IV Move
In this move, we push the high points of a surface with a saddle through a single other surface.
Type V Move
This move adds a double curve to a surface by adding a twist to the surface (in some sense, pushing part of the surface through itself)..
Type VI Move
In this move, we take a surface that has a twist in it, and "untwist" the middle portion of this twist. It may be helpful to think of it as pulling the sides of the surface (but only in the middle region) until the twist tightens down to nothing.
Type VII Move
This move simply extends a pinch through a single other surface.