Math 7590 Information

# Additional Course Materials: Math 7520: Algebraic Topology: Cohomology

In reverse chronological order:

## Hopf Algebras, Ribbon Graphs & Moduli Space references

• Good introductions to Hopf Algebras (for topologists) are Kassel's book on "Quantum Groups" & S. Shnider and S. Sternberg: Quantum groups: From Coalgebras to Drinfel'd. Algebras
• For ribbon graph: Lando/Zvonkin "Graphs on Surfaces" and last fall's course Ribbons
• Here's a set of references for Teichmuller space by David Ben Zvi: Teichmuller Space (on MathForum)
• The Margalit primer on the Mapping Class Group is here: Primer
• Here's the Penner reference: LambdaLengths
• Finally, the book by Francis Bonahon: Low-dimensional geometry: from euclidean surfaces to hyperbolic knots
• and, of course, Wikipedia.

## Hopf Fibration

The Hopf fibration is ubiquitous in topology. Its fibres give the Hopf link (& lead to linking numbers.) The stable one-stem of the stable homotopy of spheres is represented by the Hopf fibration. The references below gives some of the applications to physics and the cell structure of CP[n].
• The Hopf fibration-seven times in physics
• Physical interpretation of Hopf fibration
• Cell Structure CP[n]: McCrory
• ## Exercises

The first is a Mathematica notebook for ribbon graphs (a.k.a. dessins), the second a set of exercises on Euler characteristic, minimal triangulation of a surface and the Riemann-Hurwitz formula.

• Dessins7520.nb
• ExerciseRiemannH_0.pdf

## Differential Forms and deRham cohomology

Besides wikipedia on the above terms, the following is a good overview with further references, particularly the books of Fleming, Flanders and the omnibus: Tu/Bott. The MIT Course on Geometry of manifolds in Fall, 2004 contains an excellent introduction to manifolds with smooth atlases, vector bundles, transversality, Poincare's Lemma and a proof of de Rham's Theorem.

• Introduction to differential forms: by Donu Arapura
• MIT Course: Geometry Manifolds
• ## Cohomology of Graphs

The cohomology of graphs presents interesting features of cohomology (particularly infinite graphs.) There is a good introduction in Chapter 2 of B. Bollobas's book: Modern Graph Theory, relating the computation of the cohomology groups of a geometric graph (as a topological space) with graph theoretic notions (e.g. incidence and adjacency matrices, Kirchoff's laws, etc). An interesting related exercise is to modernize the paper below and to implement the computation in the Mathematica notebook on ribbon graphs above.
• Cohomology of Graphs: S.B. Maurer
• Comments --> Last update: 1 September , 2010