Homotopy group

Imagine you have two different shapes, like a circle and a square. You might think they are very different, but sometimes they are actually similar in a special way.

The homotopy group helps us see how similar two shapes are. It's like trying to figure out if two things are the same by looking at how they can change into each other. This is a bit like putting on different hats - you might look different, but you're still you underneath.

To find the homotopy group of a shape, we look at all the different ways we can stretch and bend it without breaking it. For example, we can take a circle and stretch it into a square by stretching out the top and bottom. We can do this because the circle and square are in the same homotopy group - we can change one into the other without any kind of cutting or gluing.

But some shapes are not in the same homotopy group. For example, we cannot change a circle into a figure eight without breaking it. This is because the figure eight is in a different homotopy group than the circle.

So, by looking at the homotopy group of a shape, we can start to see how it is related to other shapes in a very special and useful way.