Cup Product Examples

Algebraic Topology · Axiom Academy

EXAMPLE Cup Product Calculations Computing cup products on the torus, projective spaces, and seeing how they distinguish spaces Excellent work! You've explored cup products on multiple spaces. Here's what we learned: Cup product structure: The cup product turns cohomology into a graded ring, providing richer algebraic structure than homology alone. Torus example: On T² = S¹ × S¹, we have H¹(T²) = ℤ² with generators a, b satisfying a ∪ b = generator of H²(T²) ≅ ℤ, showing the torus has non-trivial multiplicative structure. Projective spaces: On ℝP² with ℤ/2 coefficients, α ∪ α ≠ 0, while on ℂP², the generator α in H²(ℂP²) satisfies α ∪ α = generator of H⁴(ℂP²). Distinguishing spaces: Cup products distinguish spaces with identical homology groups. For example, S² ∨ S⁴ and ℂP² both have H²≅ℤ and H⁴≅ℤ, but their cup product rings differ: the wedge sum has trivial cup products while ℂP² has α² generating H⁴. Ring structure matters: The cohomology ring H*(X) with cup product is a finer invariant than homology H*(X), capturing how cohomology classes multiply. Cup products are essential for understanding the multiplicative structure of cohomology and provide powerful tools for distinguishing topological spaces!

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