Cohomology

Definition

Cohomology is a fundamental invariant in algebraic topology that assigns algebraic structures (typically abelian groups or rings) to topological spaces, measuring their global properties through local data.

Cohomology groups $H^n(X;G)$ are defined for a topological space $X$, dimension $n$, and coefficient group $G$.

Construction

Cohomology can be constructed via several equivalent approaches:

• Singular cohomology: Defined using cochains on singular simplices
• Čech cohomology: Defined using open covers
• De Rham cohomology: For smooth manifolds, using differential forms
• Sheaf cohomology: A general categorical framework

Properties

• Cohomology is a contravariant functor from topological spaces to graded rings
• The cohomology ring has a multiplication structure (cup product)
• Satisfies the Eilenberg-Steenrod axioms for a cohomology theory
• Dual to homology via the universal coefficient theorem

Applications

• Distinguishing non-homeomorphic spaces
• Studying vector bundles and characteristic classes
• Obstruction theory
• Classification of fiber bundles

Related Concepts

• Homology: The dual theory
• Algebraic Topology: The field studying topological invariants
• Chain Complex: The underlying algebraic structure
• Cup Product: Multiplication in cohomology