Cohomology Ring Examples

Algebraic Topology · Axiom Academy

EXAMPLE Cohomology Ring Examples Computing and verifying cohomology ring structures of classic spaces Excellent work! You've explored fundamental cohomology ring computations. Here's what we learned: Sphere rings are simple: H*(S²) = ℤ[ω]/(ω²) shows that the cup product of any 2-dimensional class with itself vanishes, reflecting the sphere's topology. Torus rings are exterior algebras: H*(T²) with generators α, β satisfying α² = β² = 0 and αβ = -βα demonstrates the exterior algebra structure from the product structure S¹ × S¹. Projective spaces have truncated polynomial rings: H*(ℂP^n) = ℤ[x]/(x^(n+1)) with deg(x) = 2 shows that x^n generates the top cohomology, and x^(n+1) = 0 because ℂP^n is (2n)-dimensional. Verification requires checking relations: To verify a ring structure, compute cup products of generators, check they satisfy the proposed relations, and verify dimensions match the cohomology groups. Ring structure encodes topology: The multiplicative structure distinguishes spaces with identical cohomology groups as abelian groups but different topologies. These examples are fundamental in algebraic topology. Practice computing cohomology rings for products and quotients to deepen your understanding!

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