Poincaré duality

Poincaré duality is a fancy idea that helps us understand the relationship between the holes in a shape and the surface of that shape. Imagine you have a blob of silly putty. Silly putty can be squished and stretched into different shapes, but it always has the same surface area.

Now, let's imagine you took a special picture of the silly putty's surface. This picture shows you all the hills and valleys on the surface. The picture would show that there are certain places where the silly putty is curved inwards, creating a hole.

Poincaré duality tells us that there is a special relationship between these holes and the surface of the silly putty. If we count all the different types of holes (like wee tiny ones and really big ones), and then subtract that number from the topological dimension of the surface of the silly putty (which is like the number of dimensions the silly putty can move in), we will always get the same number. This is like a special equation that helps us understand the shape better.

This idea is really helpful in lots of different areas of math and science, especially when we want to count how many holes something has or how many different ways it can be shaped. So, Poincaré duality is like a special trick we can use to understand how holes on a surface relate to the surface as a whole.