Free online courseAlgebraic Topology

Course Description

Algebraic topology connects two powerful ways of thinking: the geometric intuition of shapes and spaces, and the precision of algebraic structures. In this free online course, you will learn how topologists turn complicated spaces into computable invariants, helping you reason about continuity, holes, connectivity, and deformation in a rigorous way. If you have enjoyed abstract algebra or multivariable geometry and want a framework that unifies them, algebraic topology offers that bridge while sharpening proof skills that transfer to many areas of mathematics.

You will begin with cell complexes and the idea that many spaces can be assembled from simple building blocks. From there, the course develops the central notion of homotopy: when two maps or spaces should be considered the same from a topological perspective. Deformation retracts make this concrete by showing how a space can be simplified without changing its essential structure. These ideas naturally lead to the fundamental group, a first major invariant that captures loop behavior and distinguishes spaces that may look similar at a glance.

As the theory matures, you will see how fundamental groups behave under homeomorphisms, how they interact with maps between spaces, and how powerful tools such as the Seifert–Van Kampen theorem let you compute fundamental groups by decomposing spaces into simpler pieces. Covering spaces then provide a geometric viewpoint that explains why these computations work, illuminating phenomena like deck transformations and giving a systematic method for studying spaces through their coverings.

The course then expands from loops to higher-dimensional structure through homology. By developing simplicial and singular homology and relating them, you gain a versatile invariant that measures holes in all dimensions, not just one. You will also build comfort with exact sequences, excision, and Mayer–Vietoris, techniques that turn complex topological questions into manageable algebra. Along the way, you will strengthen your ability to follow and craft proofs, translate geometry into algebra, and check arguments carefully.

Finally, the course introduces category-theoretic language and moves into cohomology, where new structure appears. Cohomology not only complements homology, it often carries additional algebraic operations that detect subtler features of spaces. Learning about the cup product and how it behaves in classic examples such as the torus gives you a glimpse of why cohomology is so effective in modern mathematics.

Whether your goal is to prepare for advanced study, deepen your understanding of algebra and geometry, or develop tools that appear in fields like geometry, topology, and mathematical physics, this course offers a coherent pathway. By the end, you will be able to interpret major theorems, compute fundamental invariants in representative cases, and approach unfamiliar spaces with a structured set of ideas for analysis.