Basic Algebraic Topology

Chapter 7 Cohomology

Cohomology in its simplest form is related to homology just the way a vector space V is related to its dual V^∗. Its traces can be found in the integral calculus of several variables, for instance, in line integrals in complex analysis. It started appearing in topology in the works of Poincaré in his duality theorem. It appeared in Alexander’s duality theorem, in the intersection theory due to Alexander and Lefschetz, de Rham’s result on smooth differential forms on manifolds and in Pontrjagin’s duality. It took several more authors and many more years to get properly established until in 1952, when the now classical book by Eilenberg and Steenrod on Foundations of Algebraic Topology appeared. The word ‘cohomology’ was invented ...