Dehn surgery

Dehn surgery is like fixing a puzzle by cutting out a piece and gluing a new one into its place. Imagine you have a puzzle of a donut and one of the pieces in the middle is missing. To fix it, you cut out another piece from the donut and glue it into the space where the missing piece was. This is like Dehn surgery.

Instead of a puzzle, we use a shape called a manifold (which is like a donut, but can have many more dimensions). Manifolds can be imperfect and have "holes" in them that we need to fill in. To do this, we use Dehn surgery.

First, we make a cut through the manifold to create a new edge, which is where we will glue the replacement piece. Then, we take out a small, tubular piece of the manifold, which creates a gap. The replacement piece is another manifold, but it needs to fit perfectly with our original manifold to fix the hole. We glue the replacement piece into the gap, and then we sew up the cut we made earlier.

The result is a new, improved manifold without any holes or gaps! This is like fixing a puzzle, but instead of using glue and cardboard, we use mathematical tools like topology and geometry to make things fit together perfectly.