Advanced Topics in Algebraic Topology

What it is about

You will learn algebraic methods for the study of smooth manifolds, in particular De‑Rham theory, cohomology and characteristic classes. By the end you will be able to understand, prove and apply the techniques for calculating examples of central theorems (e.g. Poincaré duality, Stokes, Poincaré–Hopf).

What you will be able to do

• Understanding of the De‑Rham complex and application of the Mayer‑Vietoris principle
• Familiarity with orientation, integration of differential forms and the Stokes theorem
• Knowledge and application of the Poincaré lemmas and Poincaré duality
• Understanding of bundles, Künneth and projection formulas
• Foundations of Čech cohomology
• Understanding of the Euler class, Euler characteristic and the Poincaré–Hopf index theorem
• Ability to follow proofs of central results and to apply techniques for calculation

What the module consists of

• VorlesungDelivery of theoretical content and proofs
• ÜbungDiscussion of problems to deepen and apply the methods

Teaching method

• Vorlesungen onlineIntroduction to concepts, proof techniques and classical results
• Übungssitzungen onlineDiscussion of exercise problems for deepening understanding
• ÜbungsblätterSelf-study and deepening of the treated methods
• LektüreaufgabenPreparation and basis for discussions in the sessions