Localization formula for equivariant cohomology
Localization formula for equivariant cohomology is like sharing your candy with your friends. You have a big bag of candy and you want to share it equally with your friends. But your friends are not all standing at the same place. One friend is standing on your left, one on your right, and one behind you. So, you have to move the candy to each friend's location to share it equally with them.
In the same way, the localization formula helps us share the mathematical objects called cohomology classes equally with all the groups of symmetries (equivariant groups) of a space. We have to move (localized) these mathematical objects to each group's location to share them equally.
To do this, we use a mathematical tool called the localization theorem. It tells us how to localize the cohomology classes by finding certain points (fixed points) that are not moved by any of the symmetries. We then use these fixed points to break the space into smaller pieces that are easier to handle. We call these smaller pieces the localization subsets.
Now, we apply the localization theorem to these localization subsets to share the cohomology classes equally with all the groups of symmetries. In other words, we take the cohomology class and move it from its original location to each group's location using the fixed points.
This localization formula is very useful in many areas of mathematics, especially in algebraic geometry and topology. It helps us to understand the symmetries of a space and their effects on mathematical objects, which can lead to deeper insights and better computations.
In the same way, the localization formula helps us share the mathematical objects called cohomology classes equally with all the groups of symmetries (equivariant groups) of a space. We have to move (localized) these mathematical objects to each group's location to share them equally.
To do this, we use a mathematical tool called the localization theorem. It tells us how to localize the cohomology classes by finding certain points (fixed points) that are not moved by any of the symmetries. We then use these fixed points to break the space into smaller pieces that are easier to handle. We call these smaller pieces the localization subsets.
Now, we apply the localization theorem to these localization subsets to share the cohomology classes equally with all the groups of symmetries. In other words, we take the cohomology class and move it from its original location to each group's location using the fixed points.
This localization formula is very useful in many areas of mathematics, especially in algebraic geometry and topology. It helps us to understand the symmetries of a space and their effects on mathematical objects, which can lead to deeper insights and better computations.