Free online courseAlgebraic Topology Course

Course Description

Explore the fascinating world of algebraic topology with this comprehensive online course. Dive into the foundational concepts starting with an introduction to algebraic topology, laying the groundwork for understanding the intricate structures and principles that guide this mathematical domain.

Delve into the core subjects, including the definition of singular and relative singular homology, and grasp the Eilenberg-Steenrod axioms that serve as critical touchstones in homology theory. Learn to apply these axioms to compute homologies of spheres and understand important concepts like the snake lemma and the long exact sequence axiom.

Unravel the mysteries of homotopy invariance and the 5-lemma, explore the excision and dimension axioms, and gain insights into further nuances of the Eilenberg-Steenrod axioms. The course encourages students to explore singular homology in degree zero, understand relative homology as absolute homology, and appreciate the Hurewicz isomorphism.

The Mayer-Vietoris sequences and their application for pushouts are examined in depth, along with the suspension isomorphism and computations related to real-world objects like the Klein bottle. Delve into sophisticated mathematical theorems such as the Borsuk-Ulam and Ham-Sandwich Theorems. Discover the usefulness of cellular chain complexes, their computations, and how they relate back to singular homology.

Develop a deep understanding of the Euler characteristic, lens spaces, and projective modules. Grasp the fundamental theorem of homological algebra, alongside tensor products and their computations. Obtain thorough knowledge of the Tor functor and the universal coefficient theorem, and apply the Künneth theorem and acyclic models theorem to solve complex algebraic problems.

By the end of the course, students will have gained a robust understanding of algebraic topology, equipped with tools and techniques to tackle a wide range of problems with confidence.