Saturday, June 28, 2014

De Rham Theory (part 2)

Intuitive cohomology

Continuing from last time we will now introduce the duality between forms and chains, or the Stokes theorem which is of a fundamental importance in math and physics. Recalling that represents the boundary, and d is the exterior derivative, George Stokes

proved the following identity:

We can introduce an inner product: between a cycle Ω and a cocycle ω: < Ω , ω > as the integral of ω over the domain Ω.

Stokes theorem then simply states:  < Ω , ω > = < Ω , dω > Why do we state it like this? Because the boundary of a boundary is zero: ∂∂ = 0 implies that dd = 0 as follows:

0 = < ∂∂Ω , ω > = < Ω , dω > = < Ω , dd ω >

Recall that for boundaries we have an exact sequence (which is also called a chain complex). For the usual 3D space this means:

Then we have a De Rham co-chain complex (in co-chains the arrows point in the reversed direction):

and from dd = 0 we recognize the usual identities:

·        Gradient of a curl is zero
·        Curl of a divergence is zero.

Finally we arrive at De Rham Theorem.

From chain complexes we extract the Ker/Image homology group: Hp = Zp/Bp
From cochain complexes we the Ker/Image cohomology group: Hp DR= (α| d α = 0)/( α| d α = β)

De Rham theorem tells us that the two groups are isomorphic: Hp  ~ Hp DR

This means that we can explore the topological properties of the space by looking at the solutions of differential equation on that space.

For example, an electric charge generates an electric field around it. The spatial distribution of the electric field can tell us the location of the charge. That is why the second cohomology group H2 DR can be interpreted as an electric charge! In general cohomological classes have a physical interpretation.

Next time we will venture into the related wonderful world of Hodge theory.


  1. Your presentations are interesting. I have done some presentations, live and in front of people, of related material. I focused on homotopy theory. The double slit experiment is a form of homotopy, where there are two sets of trajectories that are distinct by a topological obstruction. The measurement of which slit the particle passes through transfers the superposition into an entanglement with a needle state. Entanglements can then be a case of topology or homotopy. I am particularly interested in the case of where a quantum system entangles with a black hole.

    Cheers LC

  2. Thanks Lawrence. I want to arrive at Maxwell's equations and their relationship with gauge theory. Then I want to generalize it to Yang-Mills. There I can introduce Yang's matrix trick and relate Yang-Mills with general relativity.

    Modern math looks intimidating, but is really simple if you build the right motivation. And is unfortunate that you don't get this from most of math books.

    By the way, if you want to have guest post for your topic of interest, my blog is always open.