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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 1

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
updated 20/2/2020

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
updated 17/3/2020