Cohomology Rings — Mathematics Knowledge System

Overview

The cohomology ring of a topological space is the direct sum of its cohomology groups, equipped with a graded-commutative multiplication called the cup product. Unlike homology, this ring structure is a strictly finer invariant that distinguishes spaces with identical homology groups. It arises naturally from the diagonal map and plays a central role in characteristic classes, Poincare duality, and the computation of homotopy groups.

Intuition

Homology measures holes in a space, but two spaces can have the same homology yet be topologically different. The cup product detects how cohomology classes 'intersect' -- for manifolds, the cup product of two classes is related to the intersection of the dual homology cycles. The torus and the wedge S^1 v S^1 v S^2 have the same homology groups, but their cohomology rings differ: the torus has a non-trivial cup product in degree 1, while the wedge does not.

Formal Definition

Definition

For a topological space X, the cohomology ring H*(X; R) is the graded ring whose additive structure is the direct sum of cohomology groups, and whose multiplication is the cup product. For singular cohomology with coefficients in a ring R, the cup product of classes [phi] in H^p and [psi] in H^q is defined at the cochain level by evaluating on singular simplices.

H^*(X; R) = \bigoplus_{n \geq 0} H^n(X; R)

Cohomology ring as a graded abelian group

cohomology-ring-grading

(\phi \smile \psi)(\sigma) = \phi(\sigma|_{[v_0,\ldots,v_p]}) \cdot \psi(\sigma|_{[v_p,\ldots,v_{p+q}]})

Cup product of cochains phi and psi on a singular (p+q)-simplex sigma

cup-product-formula

\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha \quad \text{for } \alpha \in H^p, \beta \in H^q

Graded-commutativity of the cup product

graded-commutativity

Notation

Notation	Meaning
$H^*(X; R)$	Cohomology ring of X with coefficients in ring R
$\alpha \smile \beta$	Cup product of cohomology classes alpha and beta
$H^n(X; R)$	n-th cohomology group of X with coefficients in R

Theorems

Theorem 1: Graded-Commutativity of Cup Product

For classes alpha in H^p(X; R) and beta in H^q(X; R), we have alpha smile beta = (-1)^{pq} (beta smile alpha). In particular, if p is odd, then alpha smile alpha = 0 when 2 is invertible in R.

Theorem 2: Kunneth Formula for Cohomology Rings

If H*(Y; R) is free as an R-module, there is a graded ring isomorphism H*(X x Y; R) isomorphic to H*(X; R) tensor_R H*(Y; R), where the right side has the graded tensor product ring structure.

Theorem 3: Cohomology Ring of Complex Projective Space

The cohomology ring of CP^n with Z coefficients is a truncated polynomial algebra: H*(CP^n; Z) = Z[x]/(x^{n+1}), where x is the generator of H^2(CP^n; Z).

Worked Examples

1
By the Kunneth formula, H*(T^2; Z) = H*(S^1; Z) tensor H*(S^1; Z).
$H^*(T^2; \mathbb{Z}) \cong H^*(S^1; \mathbb{Z}) \otimes H^*(S^1; \mathbb{Z})$
2
H*(S^1; Z) = Z in degrees 0 and 1, generated by 1 and alpha where alpha is in H^1.
$H^*(S^1; \mathbb{Z}) = \mathbb{Z}[\alpha]/(\alpha^2), \quad |\alpha|=1$
3
Let a, b be the two degree-1 generators from each S^1 factor. Then a^2 = b^2 = 0 and a smile b generates H^2.
4
The ring is Z{1, a, b, a smile b} with a smile b = -b smile a as the generator of H^2.

✓ Answer

H*(T^2; Z) = Z[a,b]/(a^2, b^2), an exterior algebra on generators a, b of degree 1. As groups: Z in degrees 0, 2 and Z^2 in degree 1.

Practice Problems

Hardproof writing

Prove that if X is an n-dimensional manifold and alpha in H^k(X; Z) with k > n/2, then alpha smile alpha = 0.

Mediumfree response

State and apply the Kunneth formula to compute the cohomology ring of CP^1 x CP^1.

Common Mistakes

Common Mistake

Cohomology rings with the same underlying groups are isomorphic as rings.

Spaces with isomorphic cohomology groups can have non-isomorphic ring structures. The ring structure is the finer invariant. Always check cup products, not just group structures.

Common Mistake

The cup product is commutative on the nose.

The cup product is graded-commutative: alpha smile beta = (-1)^{pq} beta smile alpha. For odd-degree classes, this means alpha smile alpha = 0 (when 2 is invertible), which is a useful vanishing result.

Quiz

What property does the cup product satisfy with respect to degree?

It maps H^p x H^q to H^{p+q}It maps H^p x H^q to H^{p-q}It maps H^p x H^q to H^{max(p,q)}It maps H^p x H^q to H^{pq}

What distinguishes the cohomology ring of T^2 from that of S^1 v S^1 v S^2?

T^2 has non-trivial cup products in degree 1, while S^1 v S^1 v S^2 has all degree-1 products zeroT^2 has trivial cohomology in degree 2S^1 v S^1 v S^2 has a generator in H^1 that squares to a non-zero classT^2 has no cohomology in degree 0

Historical Background

Cohomology itself was introduced independently by Alexander and Kolmogorov in 1935-36 as a dual theory to homology. The cup product, which gives cohomology its ring structure, was defined by Alexander in 1936 and by Whitney shortly after. Hopf's theorem on the cohomology of Lie groups (1941) demonstrated the power of the ring structure, showing that the cohomology ring of a Lie group is an exterior algebra. Steenrod operations extending the ring structure to a module over the Steenrod algebra were introduced in the 1950s.

1935
Alexander and Kolmogorov independently introduce cohomology
James Alexander, Andrei Kolmogorov
1936
Alexander defines the cup product, giving cohomology a ring structure
James Alexander
1941
Hopf computes cohomology rings of Lie groups, finding exterior algebra structure
Heinz Hopf
1947
Steenrod introduces cohomology operations extending the ring structure
Norman Steenrod

Summary

The cohomology ring H*(X; R) combines all cohomology groups under a graded ring structure given by the cup product.
The cup product is graded-commutative: alpha smile beta = (-1)^{|alpha||beta|} beta smile alpha.
The ring structure is strictly finer than the group structure and can distinguish spaces with identical homology groups.
The Kunneth formula gives H*(X x Y; R) as the graded tensor product of the cohomology rings of X and Y (under freeness assumptions).
Key computations: H*(CP^n; Z) = Z[x]/(x^{n+1}), H*(RP^n; Z/2) = Z/2[x]/(x^{n+1}), H*(T^n; Z) = exterior algebra on n degree-1 generators.

References

BookHatcher, A. Algebraic Topology. Cambridge University Press, 2002.
BookBott, R. and Tu, L. Differential Forms in Algebraic Topology. Springer, 1982.
BookMay, J.P. A Concise Course in Algebraic Topology. University of Chicago Press, 1999.