S-Cobordism theorem

Okay kiddo, do you remember when we talked about manifolds and how they're like shapes that can be bent and twisted in different ways? Well, the s-cobordism theorem is all about understanding how these shapes can change from one to another.

Imagine you have two different shapes, let's say a ball and a donut. You can take the ball and squish it into the shape of the donut, right? But what if you could also do the opposite and turn the donut back into the ball? That would mean that these two shapes are sort of like "the same" in a way, because you can always change one into the other.

Now, what if you had more complicated shapes, with lots of bends and twists? It might be harder to tell if they're "the same" or not. That's where the s-cobordism theorem comes in. It helps us figure out when two shapes can be turned into each other without doing anything too crazy.

It's a bit like a puzzle. You start with one shape, and you want to turn it into another shape. The s-cobordism theorem tells you which steps you're allowed to take - like bending and twisting - and which steps you're not - like ripping or tearing.

It's really important for mathematicians because it helps us understand the structure of these shapes and how they're related to each other. But don't worry if it seems a bit confusing - even grown-up mathematicians sometimes have trouble with it!