7.4 The Künneth formula

The Künneth formula is a powerful tool for understanding the cohomology of product spaces. It connects the cohomology of a product to the cohomology of its factors, allowing us to break down complex spaces into simpler components.

This formula is closely tied to the cup product, which we've been studying in this chapter. Together, they provide a comprehensive view of the cohomology ring structure for product spaces, helping us compute and analyze more intricate topological spaces.

Künneth Formula for Cohomology

Statement and Exact Sequence

• The Künneth formula relates the cohomology of a product space to the cohomology of its factors for topological spaces X and Y
• It states that there is a short exact sequence: $0 \to \bigoplus_{p+q=n} H^p(X;R) \otimes H^q(Y;R) \to H^n(X \times Y;R) \to \bigoplus_{p+q=n-1} \text{Tor}(H^p(X;R), H^q(Y;R)) \to 0$
  • $R$ is a principal ideal domain (integers or a field)
  • $\otimes$ denotes the tensor product
• The Künneth formula holds for any coefficient ring $R$ that is a principal ideal domain
• The Tor term measures the twisting or torsion between the cohomology groups of X and Y
  • It is trivial when either $H^(X;R)$ or $H^(Y;R)$ is a free $R$-module

Naturality and Induced Maps

• The Künneth formula is natural with respect to maps between spaces
• If $f: X \to X'$ and $g: Y \to Y'$ are continuous maps, then the induced map $(f \times g)^: H^(X' \times Y';R) \to H^*(X \times Y;R)$ is compatible with the Künneth formula
  • This allows studying the behavior of cohomology under maps
  • Induced maps in cohomology can be computed using the Künneth formula

Computing Cohomology with Künneth Formula

Procedure and Simplifications

• To compute the cohomology of a product space $X \times Y$ using the Künneth formula:
  1. Know the cohomology groups of the individual spaces X and Y
  2. Compute the tensor product part $\bigoplus_{p+q=n} H^p(X;R) \otimes H^q(Y;R)$ using known cohomology groups and tensor product properties
  3. Determine if the Tor term $\bigoplus_{p+q=n-1} \text{Tor}(H^p(X;R), H^q(Y;R))$ vanishes (often trivial with field coefficients)
• If the Tor term vanishes (cohomology groups are free $R$-modules), the Künneth formula simplifies to an isomorphism: $H^n(X \times Y;R) \cong \bigoplus_{p+q=n} H^p(X;R) \otimes H^q(Y;R)$
• When the Tor term is non-trivial, compute it using the definition of Tor as the derived functor of the tensor product

Iterative Application and Examples

• The Künneth formula can be applied iteratively to compute the cohomology of a product of more than two spaces
  • Example: The n-torus $T^n$ is the product of n circles, so its cohomology ring can be computed using the Künneth formula and the known cohomology of the circle
• Other examples where the Künneth formula is useful:
  • Computing the cohomology rings of product manifolds
  • Proving results about cohomological dimension and Lusternik-Schnirelmann category of product spaces

Künneth Formula and Cup Product

Cross Product and Ring Structure

• The Künneth formula and the cup product are closely related in the cohomology of product spaces
• The cross product map $\times : H^p(X;R) \otimes H^q(Y;R) \to H^{p+q}(X \times Y;R)$, defined by $(a \times b)(x,y) = a(x) \cdot b(y)$, relates the tensor product to the cup product structure
  • It is natural with respect to maps between spaces
  • Satisfies the Leibniz rule: $(a \times b) \smile (a' \times b') = (-1)^{pq'} (a \smile a') \times (b \smile b')$
• When the Tor term vanishes, the cross product induces an isomorphism of graded rings: $H^(X;R) \otimes H^(Y;R) \cong H^*(X \times Y;R)$
  • The tensor product is equipped with the multiplication $(a \otimes b) \cdot (a' \otimes b') = (-1)^{pq'} (a \smile a') \otimes (b \smile b')$
• In this case, the cup product structure of $H^(X \times Y;R)$ is completely determined by the cup product structures of $H^(X;R)$ and $H^*(Y;R)$ via the Künneth isomorphism and the cross product

Relationship and Computational Implications

• The Künneth formula and the cup product are interconnected in the cohomology of product spaces
• Understanding their relationship helps in computing and understanding the ring structure of the cohomology of product spaces
• The cross product provides a way to relate the tensor product in the Künneth formula to the cup product structure
• When the Tor term vanishes, the cup product structure is fully determined by the factors' cup product structures via the Künneth isomorphism and cross product

Applying the Künneth Formula

Problem-Solving Strategies

• The Künneth formula is a powerful tool for computing the cohomology of product spaces in terms of the cohomology of the factors
• When solving problems, consider:
  • The coefficient ring $R$ and whether the Tor term vanishes (simplifies computations)
  • Combining with other tools like the Mayer-Vietoris sequence and spectral sequences for more complicated spaces
• The naturality of the Künneth formula allows studying the behavior of cohomology under maps and computing induced maps in cohomology

Applications and Examples

• Computing cohomology rings of tori and other product manifolds
  • Example: The n-torus $T^n$ cohomology ring via the Künneth formula and the known cohomology of the circle
• Proving general results about cohomology of product spaces
  • Cohomological dimension
  • Lusternik-Schnirelmann category
• Decomposing complicated spaces into products or fibrations
  • Combining the Künneth formula with the Mayer-Vietoris sequence and the spectral sequence of a fibration to compute cohomology
• Studying the behavior of cohomology under maps and computing induced maps in cohomology using the naturality of the Künneth formula