Cohomology

Lore & Background

Cohomology is a sequence of abelian groups often defined from a cochain complex. It can be viewed as a method of assigning richer algebraic invariants to a space than homology, and some versions arise by dualizing the construction of homology. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications, related to functions and pullbacks in geometric situations: given spaces X and Y, and some function F on Y, for any mapping f: X → Y, composition with F gives rise to a function F ∘ f on X.

A simple example is the de Rham cohomology of the circle. Smooth functions on the circle can be thought of as periodic functions f: ℝ → ℝ with f(θ+2π)=f(θ). The differential of a function f(θ) is df(θ)=f'(θ)dθ. Differentials of a periodic function have the property that their integral over a whole period is zero. A differential dθ by itself is inexact, as its integral over a period is 2π. Any differential form g(θ)dθ is inexact precisely when its integral over a period is nonzero, and exact when that integral is zero. The quotient space of inexact differentials modulo exact differentials is one-dimensional.

Reader's Guide

Cohomology became a dominant method in the mathematics of the second half of the twentieth century, spreading from topology throughout geometry and algebra. The most important cohomology theories have a product, the cup product, which gives them a ring structure, making cohomology usually a stronger invariant than homology. The common pattern across cohomology constructions is that one has objects called cochains, a coboundary operator measuring the failure of a cochain to satisfy a compatibility condition, and cohomology classes obtained by identifying cochains that differ by a trivial or exact contribution. The Möbius strip provides an illustrative example: it is not the product space of a line segment with a circle (a cylinder), but locally resembles an ordinary rectangle. The obstruction to making the product structure global is encoded in the first cohomology of the underlying circle, H¹(S¹, {+1,−1}), which classifies the two inequivalent ways a line can twist around a circle (even or odd number of twists). This group is isomorphic to {+1,−1}, where +1 corresponds to an even number of twists and −1 to an odd number.

Did You Know?

• The Möbius strip's obstruction to being a global product is encoded in the first Stiefel-Whitney class w₁ in H¹(S¹, ℤ/2ℤ).
• Cohomology is a contravariant theory, which is more natural than homology in many applications due to functions and pullbacks.