The Cohomology Ring

Algebraic Topology · Axiom Academy

Understanding the graded ring structure on cohomology and its power as a topological invariant 1. The Cohomology Ring H*(X; R) The cohomology ring is the direct sum of all cohomology groups, equipped with the cup product: The cup product is a bilinear map that combines cohomology classes from different dimensions: A graded ring R = ⊕ R n satisfies: Addition: within each graded component Multiplication: R p × R q → R p+q Graded commutativity: α ⌣ β = (−1) pq β ⌣ α for α ∈ H p , β ∈ H q 2. Computing Cohomology Rings of Simple Spaces Let's examine cohomology rings of fundamental spaces to build intuition: For a point space pt with coefficients in R: This is the simplest possible ring: just the coefficient ring R in degree 0. The cohomology ring of the circle (with ℤ coefficients): where α ∈ H 1 (S 1 ). The ring structure: α ⌣ α = 0 (cup square vanishes in odd degree). 3. H*(S n ) as a Truncated Polynomial Ring The n-sphere has a particularly elegant cohomology ring structure: where ω is a generator of H n (S n ) ≅ ℤ. This is an exterior algebra on one generator. The key relation is ω ⌣ ω = 0, which follows from dimensional analysis: ω ∈ H n (S n ), so ω ⌣ ω would lie in H 2n (S n ) But H 2n (S n ) = 0 for n > 0, forcing the relation We write H*(S n ) ≅ ℤ[ω]/(ω 2 ) - polynomial ring truncated at degree 2 With ℤ/2ℤ coefficients (to avoid torsion complications): where α ∈ H 1 (ℝP n ; ℤ/2ℤ). The powers α k = α ⌣ ··· ⌣ α generate H k for k ≤ n, and α n+1 = 0.

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