Simplicial homology

Okay, so let me first ask you if you know what a shape is? You know, like a square or a circle? Great! Now imagine some shapes that are made up of line segments, like triangles or rectangles. These shapes are called simplicial complexes.

Now, imagine you have a very special set of instructions on how to take any given simplicial complex and turn it into some numbers. These numbers describe the shape in a special way, and we call them the simplicial homology of the shape.

Basically, we start by counting how many little pieces the shape is made up of. These pieces are called simplices, and they can be points, lines, triangles, or any other simple shape. Then we look at how these simplices are connected.

For example, imagine a triangle. We would say that it has 3 vertices (the points where the lines meet), 3 edges (the lines connecting the vertices), and 1 face (the whole triangle).

Now, imagine we had a square. We would say that it has 4 vertices, 4 edges, and 1 face. But if we added another line to cut the square in half, we would end up with two triangles. Now our shape would have 4 vertices, 5 edges, and 2 faces.

From these numbers, we can figure out more information about the shape. The simplicial homology of a shape tells us how many loops or holes it has, and where they are.

So, if you imagine a rubber band stretched out into a circle, it has one loop. That means its simplicial homology has one "hole". If we took that rubber band and tied a knot in it, we would have two loops. That means its simplicial homology has two "holes".

By understanding the simplicial homology of a shape, we can learn more about its properties, like how easy it is to stretch or bend without tearing it apart. And that's it, you've learned about simplicial homology!