Regular homotopy

Okay kiddo, let's talk about regular homotopy. Have you ever seen a piece of rubber band? When you pull it and twist it around, it looks different than when it's just lying flat on the table, right? But even though it looks different, it's still made of the same material and can be pulled and twisted back into its original shape.

Now let's imagine we have two different shapes made out of rubber bands. They might look really different from each other, but we want to know if they could be twisted and pulled to look exactly the same. That's called regular homotopy.

To see if two shapes can be regularly homotoped, we take one of the shapes and imagine we're stretching and bending and twisting it around. We have to make sure that we're only doing certain kinds of moves, though. We can't just cut the rubber band or poke a hole in it or anything like that. We can only do moves that keep the rubber band connected and don't change its overall shape.

If we can stretch and bend and twist the first rubber band into the second rubber band using those kinds of moves, then we say the two shapes are regularly homotopic. It's like they're part of the same family, even if they look different from each other.

So that's regular homotopy! It's a way of figuring out if two shapes can be twisted and bent around to look the same.