Cohomology Summary

Algebraic Topology · Axiom Academy

Key concepts and computational techniques in cohomology theory. Cochain Complexes & Cohomology Groups Cochain Complex: The dual construction to chain complexes, with coboundary maps where Cocycles & Coboundaries: are closed forms (in the kernel), are exact forms (in the image) Cohomology Groups: Defined as , measuring "holes" from a dual perspective Why Dual?: Cohomology assigns values to chains rather than being built from points, giving algebraic rather than geometric structure Cup Product: A multiplication operation making cohomology into a graded ring Graded Commutativity: Satisfies where |α| is the degree of α Richer Structure: Unlike homology, cohomology has both addition and multiplication, revealing algebraic invariants Applications: The ring structure distinguishes spaces that homology alone cannot (e.g., ) Computing Cohomology: The Torus Start with Homology: For the torus , we know Apply Universal Coefficient Theorem: Since Z is free, , giving Compute Cup Products: Let α, β generate . Then generates Cohomology Ring: The full ring structure is , where Poincaré Duality for Manifolds Statement: For a closed, oriented n -manifold M , there's an isomorphism Geometric Meaning: k -dimensional cycles are dual to (n-k) -dimensional cycles, relating cohomology classes to homology classes Cap Product: The isomorphism comes from capping with the fundamental class Example: For surfaces of genus g , Poincare duality relates de Rham Cohomology & Differential Forms

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