This is a short set of notes that covers a couple of aspects of duality in differential geometry and algebraic topology. It grew out of an enigmatic comment I encountered, to the effect that the Lie and exterior derivatives were almost-dual in some sense. I wanted to ferret out what this meant, which turned out to be more involved than anticipated. Along the way, I decided to explore something else I never had properly understood: the nature of integration from a topological perspective. This led to an exploration of the equivalence of de Rham and singular cohomology.
The notes are in the form of five sets of slides. Originally, they comprised four presentations I gave in a math study group. On tidying, the last set grew unwieldy, so I broke it into two.
- Lecture1: Review of DG and AT. Types of derivatives on , de Rham Complex, review of some diff geom, Lie deriv and bracket, chain complexes, chain maps, homology, cochain complexes, cohomology, tie in to cat theory.
- Lecture2: The integral as a map, Stokes’ thm, de Rham’s thm, more about Lie derivs.
- Lecture3: Recap of de Rham cohomology, review of relevant algebra, graded algebras, tensor algebra, exterior algebra, derivations, uniqueness results for derivations, the interior product.
- Lecture4: Cartan’s formula, tensor vs direct product, element-free def of LA, Lie coalgebras
- Lecture5: Quick recap, relation between struct constants of LA and LCA, the choice of ground ring or field, duality of Lie deriv and exterior deriv.
These notes grew organically, so the order of presentation may seem a bit … unplanned. The emphases and digressions reflect issues I encountered, and may be peculiar to my own learning process and the many gaps in my physicist-trained math background. Others may not share the same points of confusion, or require the same background explanations. They were designed for my own use at some future point when I’ve completely forgotten the material and need a bespoke refresher. I.e., a week from now.
Although I’ve tried to polish the notes to stand on their own, there are some allusions to earlier material studied in the group. In particular, certain abbreviations are used. Here is a (hopefully) complete list:
- DG: Differential Geometry
- AT: Algebraic Topology
- DR: de Rham
- : Used for a Principal bundle. Not really used here, but mentioned in passing.
- PB: Principal Bundle. Not really used here, but mentioned in passing.
- AB: Associated Bundle. Not really used here, but mentioned in passing.
- LG: Lie Group. Mentioned in passing.
- LA: Lie Algebra
- LCA: Lie Coalgebra (defined here).
- v.f. Vector fields
- v.s. Vector space
The 1st 2 lectures focus on the equivalence of de Rham and singular cohomologies via a duality embodied in the integral map, and enforced by Stokes’ and de Rham’s thms. The last 3 lectures focus on the quasi-duality between the Lie derivative and exterior derivative. By quasi-duality we don’t mean to downplay its legitimacy. I didn’t go through all sorts of contortions to call a square a circle just because it sounds elegant. There is a true duality, and a beautiful one. But saying that it is directly between the Lie and exterior derivs is slightly misleading.
These notes were constructed over a period of time, and focus on the specific topic of interest. They are by no means comprehensive. Although edited to correct earlier misconceptions based on later understanding (as well as errors pointed out by the math group), the order of development has not been changed. They were written by someone learning the subject matter as he learned it. They may have some mistakes, there may be some repetition of points, and they are not designed from the ground up with a clear vision. Nonetheless, they may prove helpful in clarifying certain points or as a springboard for further study.
These notes explain the following:
- as a map from the de Rham complex to the singular cochain complex
- Stokes’ thm as a relationship between de Rham cohomology and singular cohomology
- The various types of derivations/anti-derivations encountered in differential geometry
- A review of graded algebras, tensor algebras, exterior algebras, derivations, and anti-derivations.
- A review of Lie Derivatives, as well as Cartan’s formula
- A discussion of what the duality of and means
- A discussion of the two views one can take of and : as -dimensional vector spaces over or as finite-basis modules over the smooth fns on M. The former is useful for abstract formulation while the latter is what we calculate with in DG. The transition between the two can be a source of confusion.
- A discussion of why derivations and anti-derivations are the analogues of linearity when we move from one view to the other.
The notes draw from many sources, including Bott & Tu, Kobyashi & Nomizu, and various discussions on stackexchange. A list of references is included at the end of the last set of slides.