Physics and Geometry

What exactly is cohomology?

More than a year ago I started this blog as an experiment, wondering if I would have enough interesting things to say. I think I ran out of simple interesting topics, and it is time to take the level up a notch. After all, how many interesting posts about strange correlations and coloring games in quantum mechanics can one do? Fortunately the quantum mechanics (and physics in general) is huge, but we need to understand its geometric language. There are three large mathematical mountains blocking our path to understanding physics the modern way: homotopy, homology, and cohomology theory. I will attempt to guide you through this terrain and explain it in as much intuitive terms as I can. Once we conquer this area, a vast and interesting landscape waits for us. Then we should be able to explain electromagnetism, Yang-Mills gauge theory, and the Standard Model. My end goal is to make the following statement self-evident: fermions are sections and bosons are connections in a vector bundle.

For the first couple of weeks I will start with simple geometrical topics, and after I will attend an upcoming physics conference, I will stop the geometry posts and present impressions from the conference. Then I will resume the quest to explain the beautiful area of differential and algebraic geometry.

Today’s topic will be: Euler’s characteristic

After the discovery of Euclidean geometry, nothing happened in geometry for a long time, until merchants sailing the oceans needed accurate maps. This forced introduce the idea of coordinates and distance. Oddly enough, ancient Greeks taught geometrically using constructions with ruler and compass, but they lacked the idea of coordinates. Once coordinates are introduced, a discrete mathematical structure is superimposed on a continuous domain, and essential information about geometry is encapsulated in algebraic notions. Geometry is like playing the violin, while algebra is like playing the piano. Each piano key makes a distinct repeatable sound and is easier to play complex songs because you don’t have to worry about accurately recreating the notes. In the same way, algebraic methods are much more powerful than simply visualizing potentially highly complex geometrical spaces.   

So let us start with a deep and apparent trivial observation. Draw a triangle on a piece of paper. Pick a point outside the triangle and connect it with the closest side of the triangle. You just drew two additional lines, created a new vertex, and a new face. If you repeat the process, the number of FACES+VERTEX-SIDES (F+V-S) stays the same. For the original triangle F+V-S = 1+3-3 = 1

Suppose you place the additional point inside of a triangle: then you add 2 new faces, 1 vertex, and 3 sides and F+V-S continues to stay 1 regardless of where you add the new point, inside or outside of the triangle. 

So what is the big deal? The big deal is that if you start with a triangle and add a point in space above the original plane then F+V-S changes. For a tetrahedron this number is now 2 because when you add a point in space, you create 3 new faces instead of 2 and F+V-S is able to tell distinguish between plane figures and geometrical figures in space.

But can't we simply just see that a geometric objects is in space and not in a plane? Sure, but this works because we embed the object into the ordinary 3D space. If we are talking about crazy complex 26-dimesional objects for example we lack the intuitive embedding and we need a way to use intrinsic object characteristics to be able to say something meaningful.

The key point is that F+V-S which is called Euler’s characteristic is an intrinsic geometrical invariant. To be continued…