Manifold decomposition

A manifold is like a shape, but it's a special kind of shape that looks the same at every point. It's sort of like a ball or a donut. But sometimes, people want to take a really complicated manifold and break it down into simpler pieces. This is called manifold decomposition.

It's kind of like taking a big puzzle and breaking it up into smaller puzzles that are easier to solve. Once you've solved all the small puzzles, you can put them back together and solve the big puzzle.

When we talk about a manifold, we often talk about "cohomology." Cohomology is like a way of measuring how complicated a manifold is. The more "holes" or "twists" a manifold has, the more complicated it is. But just like with a puzzle, it's easier to solve if you break it up into smaller pieces. That's what manifold decomposition does - it breaks up a complicated manifold into simpler pieces that we can understand.

People use manifold decomposition for all kinds of things. It helps them understand the shape of the universe, or how different particles interact with each other. It's a really useful tool for scientists who study some of the most complicated aspects of our world!