What is Cohomology?

Algebraic Topology · Axiom Academy

Discover the dual perspective that transforms how we measure topological spaces. Step 1: What Homology Tells Us Let's start by recalling what homology does. Explore different spaces and see what holes homology detects. Step 2: Flipping the Perspective Instead of counting chains directly, what if we measure functions on chains? Click on chains to assign them values. Step 3: What Does Cohomology Detect? Adjust the slider to see how cohomology reveals the same topological features from a different angle. Step 4: Extra Structure - The Cup Product This is where cohomology truly shines! Click on the boxes to explore how cochains can be multiplied. Cohomology groups are dual to homology groups (often isomorphic for nice spaces), so they detect the same holes and features. But cohomology comes equipped with the cup product. The cohomology ring H^*(X) has both addition and multiplication (cup product). This ring structure encodes subtle information about how different dimensional holes interact. Cohomology is often easier to compute using algebraic techniques. Functors that are contravariant on chains become covariant on cochains, making many constructions more natural. The cup product enables fundamental results like Poincaré duality, the Künneth formula, and characteristic classes. These tools are central to modern topology and geometry.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.