Introduction to graph complexes
Graph complexes are chain complexes with very simple descriptions in terms of finite graphs. They were introduced by Kontsevich in his work on deformation quantization, in particular in the proof of his “formality theorem,” but have proved to have applications in a wide variety of other areas. These include the study of groups important in low-dimensional topology such as automorphism groups of free groups and surface mapping class groups. This course will be an introduction to these complexes, with special emphasis on their applications in geometric group theory.
Exercises (updated 17/3/2020)
Lecture | Topics | Edited Notes |
---|---|---|
Lecture 1 | Introduction, oriented graphs, (cyclic) operads O | |
Lecture 2 |
O-graphs, Lie algebra generated by O-spiders, Chevalley-Eilenberg homology of a Lie algebra, statement of Kontsevich's theorem |
Lecture 2 updated 20/2/2020 |
Lecture 3 | Proof of first statement of Kontsevich's theorem | Lecture 3 |
Lecture 4 | Recap, moduli spaces of graphs and commutative graph homology |
Lecture 4 |
Lecture 5 |
Statement of Willwacher's theorem, Lie graphs and forested graphs, spine of Outer space |
Lecture 5 updated 20/2/2020 |
Lecture 6 | Cube complex structure of spine of Outer space, identifying cohomology of spine with homology of forested graph complex | Lecture 6 |
Lecture 7 | Homotopy type of the complex of trees, Associative graph homology and mapping class groups | Lecture 7 |
Lecture 8 | Even commutative graph homology: relation with mapping class groups of closed surfaces, Lie algebra structure |
Lecture 8 |