Algebraic Topology Course by Roman Sauer

About this course

Algebraic topology is where geometry meets algebra: you translate the features of spaces into algebraic objects that can be computed and compared. This free online course guides you through that translation with singular homology as the central tool, helping you move from pictures and intuition to rigorous, reusable methods. If you want a clearer way to distinguish spaces, detect holes, and understand when two shapes are essentially the same, this course gives you the framework to do it with confidence.

You start by building the language of singular homology from the ground up, including the role of simplices and the chain-level viewpoint that turns continuous problems into algebra. From there, the course develops relative homology so you can analyze a space together with a subspace, a perspective that becomes indispensable in real computations and in proofs that compare constructions. Along the way, you learn why axioms matter: the Eilenberg-Steenrod approach shows which properties characterize a homology theory and how these properties unlock results such as the homology of spheres and functorial behavior under maps.

A major theme is learning to navigate the machinery that makes homology powerful rather than mysterious. You work with induced maps, exactness, and core homological tools such as the snake lemma and the 5-lemma, gaining the ability to pass information through diagrams and conclude isomorphisms when direct computation would be difficult. Key structural principles like homotopy invariance clarify which features of a space are truly topological, while excision and the small simplices perspective explain why local modifications often do not change homology, paving the way for practical decompositions.

As your understanding matures, the course emphasizes computational strategy and conceptual connections. Degree zero homology provides a crisp algebraic lens on connectedness, while the Hurewicz viewpoint links homology with more geometric invariants and offers intuition for what homology is measuring. You also meet the triple sequence and see how relative homology can be reframed as absolute homology in suitable settings, improving flexibility when you tackle examples.

To tie everything together, Mayer-Vietoris becomes a central method: it shows how to compute the homology of a complicated space from simpler overlapping pieces, including versions suited to pushouts. By the end, you will be better equipped to read and write proofs in algebraic topology, understand standard theorems in a principled way, and approach new spaces with a reliable toolbox for breaking them apart and extracting invariant information.