Atiyah–Hirzebruch spectral sequence
So imagine you have a bunch of pictures of different shapes like circles, squares, and triangles. Now let's say you want to count how many corners are in all the pictures put together. But it's really hard to just look at all the pictures and count them all by hand.
That's where the Atiyah-Hirzebruch spectral sequence comes in. It's like a special way to count the corners in all the pictures without having to look at every single picture one by one.
To do this, we make a big table with a bunch of different rows and columns. Each row is a different type of shape (like circles, squares, and triangles), and each column is a different number of corners (like 0, 1, 2, 3, etc.).
Then, we start looking at the pictures one by one and figuring out which row and column it should go in. For example, a circle has 0 corners so we would put it in the row for circles and in the column for 0 corners.
Once we have all the pictures sorted into the table, we can start counting the corners. We start with the pictures in the row for circles and add up all the numbers in the columns. This tells us how many corners all the circles have combined. Then we move on to the squares row and add up all the corners for the squares, and so on.
So instead of having to look at every single picture and count its corners, we can use the Atiyah-Hirzebruch spectral sequence to organize all the pictures and count the corners more efficiently.
That's where the Atiyah-Hirzebruch spectral sequence comes in. It's like a special way to count the corners in all the pictures without having to look at every single picture one by one.
To do this, we make a big table with a bunch of different rows and columns. Each row is a different type of shape (like circles, squares, and triangles), and each column is a different number of corners (like 0, 1, 2, 3, etc.).
Then, we start looking at the pictures one by one and figuring out which row and column it should go in. For example, a circle has 0 corners so we would put it in the row for circles and in the column for 0 corners.
Once we have all the pictures sorted into the table, we can start counting the corners. We start with the pictures in the row for circles and add up all the numbers in the columns. This tells us how many corners all the circles have combined. Then we move on to the squares row and add up all the corners for the squares, and so on.
So instead of having to look at every single picture and count its corners, we can use the Atiyah-Hirzebruch spectral sequence to organize all the pictures and count the corners more efficiently.