Cohomology | Algebraic Topology Lecture 23 | Nge Kie Seng 251218

This lecture introduces cohomology in algebraic topology, starting with motivation from vector calculus. The instructor recalls the concept of conservative vector fields and the condition for a vector field to be conservative on a simply connected domain. He then shows that this condition fails on non-simply connected domains, such as the punctured plane, where the vector field (rotation divided by radius) has equal mixed partials but is not a gradient. This motivates the definition of the first cohomology group as the quotient of closed forms by exact forms. The lecture extends this to higher dimensions using differential forms and the exterior derivative, leading to de Rham cohomology. The instructor also draws an analogy with elementary arithmetic, showing how carrying in addition can be seen as a cocycle condition. Finally, he discusses the categorical perspective, mentioning natural transformations and the contravariance of the cohomology functor.