The following ultra-short intro to (co)homology is a reaction to this recent article which appeared on quantamagazine here:
Thanks to John Wehrle for sharing, and to Andreas Geisler for making me think about it:
In homology, one defines a boundary operator \(\delta\) which associates to each n-dimensional shape a number of n-1 dimensional shapes. If I grab a filled-in triangle (a 2-simplex) then \(\delta\) will get me the complex consisting of its three edges (1-simplices) glued together to an empty triangle.
The fundamental insight here is that “the boundary of a boundary is empty”. For example, the boundary of an empty triangle (or any loop) is the empty set. That’s because the boundary of a line (1-simplex) consists of a disconnected pair of points (0-simplices), and by gluing a boundary on top of another boundary of matching dimension, we remove both boundaries. By forming a loop we end up with no boundary at all.
The fun thing with homology is that you can interpret \(\delta\) as a linear operator sending all points in the interior of a simplex to its boundary. You can then write down our fundamental insight as equation \(\delta \delta x = 0\) and then solve it! So there’s a direct way to get from our topological description to doing the same with linear spaces! Homology generalizes like crazy!
Well, actually, it’s a bit annoying that \(\delta\) returns lots of n-1 simplices for any n-simplex: it is a multifunction. If we instead map the other way around, from the boundary to the inside, we simply get a function! And that’s called cohomology, and is what the cool kids do all the time!
Extra: Homology is so cool! It comes in many forms and is related to other cool stuff. There’s de Rham cohomology based on differential forms, or Hodge theory for differential equations, one can go about with Morse theory, which computes homologies by looking at movies of slices of manifolds, and many more variants exist. Homology is basically a generalization of the Euler characteristic, and like it, relates to Betti numbers. And they have relations to all sorts of other stuff.
One can compute the cohomology of a cohomology, which yields number carpets. And this is a nice starting point for Adam’s spectral sequence, a method to compute homotopy groups with cohomology. Homotopy is that other, older method to study holes in manifolds: it gives groups which describe the ways in which one can wrap loops or spheres around holes of various dimension. Homology was invented because homotopy groups are so incredibly difficult to compute. They still are, we’re not there yet!
Oh, right, References, always add some further reading:
I liked Allen Hatcher’s nice, free textbook “Algebraic Toplogy” which explains how to compute homology for simplical complexes, which are basically simplices of various dimension glued together. You can find it here:
It is a textbook, which means it’s deep enough to put some work into it. It also does a bit more than simplical homology. Just so you don’t feel overwhelmed, it’s okay to read only part of it, and it’s worth giving it multiple passes!
I also enjoyed John Milnor’s classic “Morse Theory” for it gave a different perspective on homology. You can get download it for free here:
Also a textbook, but it states the basic intuitions clearly.
I have no good book for de Rham cohomology, but if you have basic knowledge about homology (like, after reading just a little of Hatcher’s book), and get what differential forms are, say from watching parts 3,4,5 of Keenan Crane’s lectures on differential geometry here:
You might be able to cope with what Wikipedia has on de Rham cohomology.
Generally, what Wikipedia has on all of these buzzwords above isn’t too shabby either. For building intuition it’s always great to have alternative viewpoints available.
And that's what I did for the rest, too. Suggestions, anyone?
I have Adams' book on his e-invariant lying around here somewhere, but haven't read it yet:
But I've read some on the nLab here:
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