Last post Anna talked about representing a large tensor by a collection of smaller ones, which gave me the inspiration to write this post.
A lot of the time in topology (or at least in the papers I read) it’s convenient to use a tool called a spectral sequence to arrive at a result. In general, the spectral sequence breaks down the computation you want to perform in to a lot of smaller computations. This can seem like more work but in truth it can sometimes be impossible to perform your original computation in any other way.
A spectral sequence can be thought of as a book with a grid on each page, as shown above. On each vertex of the grid lies a group (shown as black dots in the picture) and there are maps going between certain groups (shown as arrows in the picture) satisfying certain properties. Given the first page there is a formula to give you the second page, and so on. Therefore if this was a book, the first page would determine the whole story. But unlike a story, this book need never end. If a group at a certain grid point stabilises (stays the same from a certain page forevermore), then the group at that grid point is called the limiting group at that point, and these are the groups that can give us answers to computations.
In particular you can use a spectral sequence to calculate the homology of a space, using only knowledge about a filtration of the space: a string of subspaces of that space which fit inside each other. So if you had a space built from dots, lines, triangles etc. then you can calculate it’s homology using knowledge of the interactions between the space with only the dots, only the lines, only the triangles etc. (it’s actually something called relative homology but we wont get into that). Quite cool eh? For example
Can be broken up into these three spaces, each one fitting inside the next:
(I just want to mention here that this space has very boring homology and in reality the spaces and filtrations we use are a lot more complicated!)
I’ll leave you with a picture of the first page of a spectral sequence that I drew on my blackboard today.