Advanced Topics in Algebraic Topology SS 2020



Time and place:

Wed 10:15 -- 12:00

Both lectures and exercises will be given in a live stream, and will include reading assignments to be discussed in class.

Due to the current situation with Covid-19, the lectures will be held via online stream. Read the following information HERE about online lectures before joining the class. Please register for the class on TUMonline. The link to the lectures is provided in moodle and has also been sent to registered participants via TUMonline. We will discuss the logistics on the class on Wednesday, April 22. If you cannot join then, or have trouble registering for the course, please contact me via email. 


Algebraic topology is the study of topological spaces with algebraic methods. In this course we will focus on the study of smooth manifolds with algebraic tools. We will also use tools relevant to algebraic geometry and will prove some classic results in algebraic topology. Topics will include:
- de Rahm complex of manifolds, Mayer-Vietoris, sheaves
- orientation and integration of differential forms, Stokes' Theorem, Poincaré lemma
- Poincaré duality
- bundles, Künneth formula, projection formula
- Cech cohomology
- Euler class and Euler characteristic, Poincaré-Hopf index formula


The main reference for this class, which we will follow closely is:
Bott and Tu, Differential Forms in Algebraic Topology

Further references:
Madsen and Tornehave, From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes
Fulton, Algebraic Topology
Milnor, Topology from the Differentiable Viewpoint
Hatcher, Algebraic Topology
Lee, Introduction to smooth manifolds

Further reading:
Terence Tao, Differential forms and integration
Robbin and Salamon, Introduction to Differential Topology These are (incomplete) lecture notes.

Reading assignment

Exercise classes

Exercise class:   Friday 11-12; given by Tashi Walde