Homotopy groups of spheres
Homotopy groups of spheres are a way to describe how different surfaces or shapes can be twisted or bent in space without tearing or cutting them. Imagine you have a ball and a rubber band. If you stretch the rubber band around the ball, you can move it around in all sorts of ways without breaking the band or the ball.
Homotopy groups are like different levels of stretching that you can do with the rubber band. Each level of stretching creates a different homotopy group. For example, if you stretch the rubber band around the ball so that it goes around it exactly once, you create a homotopy group called the first homotopy group of the sphere.
Different homotopy groups can tell you different things about the shape or surface that you're studying. They can help you understand how many holes or knots the shape has, whether it can be stretched into a different shape, or whether there are any strange or interesting ways that it can be twisted or turned.
Overall, homotopy groups of spheres are a really cool and useful tool in math and science for describing and understanding the shapes and structures that make up our world!
Homotopy groups are like different levels of stretching that you can do with the rubber band. Each level of stretching creates a different homotopy group. For example, if you stretch the rubber band around the ball so that it goes around it exactly once, you create a homotopy group called the first homotopy group of the sphere.
Different homotopy groups can tell you different things about the shape or surface that you're studying. They can help you understand how many holes or knots the shape has, whether it can be stretched into a different shape, or whether there are any strange or interesting ways that it can be twisted or turned.
Overall, homotopy groups of spheres are a really cool and useful tool in math and science for describing and understanding the shapes and structures that make up our world!