5.4 Künneth formula

The Künneth formula is a key tool in cohomology theory, connecting the cohomology of a product space to its factors. It uses tensor products and spectral sequences to express the cohomology of a product in terms of its components.

This formula has wide-ranging applications in algebraic topology and geometry. It simplifies calculations for product spaces like tori and projective spaces, and provides insights into cohomology ring structures and cross products.

Künneth formula for cohomology

• The Künneth formula is a powerful tool in cohomology theory that relates the cohomology of a product space to the cohomology of its factors
• It provides a way to compute the cohomology of a product space in terms of the cohomology of its individual components
• The formula involves tensor products of cochain complexes and a spectral sequence argument

Tensor products of cochain complexes

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• Given two cochain complexes $C^\bullet$ and $D^\bullet$, their tensor product $C^\bullet \otimes D^\bullet$ is a new cochain complex
• The differential on the tensor product is defined using the Leibniz rule: $d(c \otimes d) = dc \otimes d + (-1)^{\deg c} c \otimes dd$
• The cohomology of the tensor product complex is related to the cohomology of the individual complexes via the Künneth formula

Künneth formula statement

• The Künneth formula states that there is a short exact sequence: $0 \to \bigoplus_{p+q=n} H^p(C^\bullet) \otimes H^q(D^\bullet) \to H^n(C^\bullet \otimes D^\bullet) \to \bigoplus_{p+q=n+1} \operatorname{Tor}(H^p(C^\bullet), H^q(D^\bullet)) \to 0$
• The direct sum runs over all pairs of degrees $p$ and $q$ such that $p+q=n$
• The $\operatorname{Tor}$ term measures the deviation from the tensor product being exact

Künneth formula proof outline

• The proof of the Künneth formula involves constructing a spectral sequence from a double complex
• The double complex is built from the tensor product of the two cochain complexes $C^\bullet$ and $D^\bullet$
• The spectral sequence converges to the cohomology of the total complex, which is isomorphic to $H^*(C^\bullet \otimes D^\bullet)$
• By analyzing the $E_2$ page of the spectral sequence, one obtains the Künneth formula

Künneth formula for topological spaces

• When applied to the singular cochain complexes of two topological spaces $X$ and $Y$, the Künneth formula relates the cohomology of the product space $X \times Y$ to the cohomology of $X$ and $Y$
• In this context, the Künneth formula takes the form: $H^n(X \times Y) \cong \bigoplus_{p+q=n} H^p(X) \otimes H^q(Y) \oplus \bigoplus_{p+q=n+1} \operatorname{Tor}(H^p(X), H^q(Y))$
• The isomorphism holds with coefficients in a field or when one of the spaces has torsion-free cohomology

Applications of Künneth formula

• The Künneth formula has numerous applications in algebraic topology and geometry
• It allows for the computation of cohomology groups of product spaces, which often arise naturally
• The formula also provides insight into the ring structure of cohomology and its relation to cross products

Cohomology of product spaces

• The Künneth formula is particularly useful for computing the cohomology of product spaces such as tori, projective spaces, and manifolds
• For example, the cohomology of the torus $T = S^1 \times S^1$ can be computed using the Künneth formula and the known cohomology of the circle $S^1$
• The formula splits the cohomology of the product into tensor products of the cohomology of the factors, simplifying computations

Cohomology ring structure

• The Künneth formula is compatible with the cup product structure on cohomology
• It allows for the determination of the cohomology ring structure of a product space in terms of the cohomology rings of its factors
• The cross product map between cohomology groups of factors induces a ring homomorphism into the cohomology of the product space

Künneth formula vs cross product

• The Künneth formula and the cross product are closely related but distinct concepts
• The cross product is a map $H^p(X) \otimes H^q(Y) \to H^{p+q}(X \times Y)$ that sends cohomology classes of the factors to a cohomology class of the product
• The Künneth formula, on the other hand, describes the full cohomology of the product space as a direct sum of tensor products and torsion products
• The cross product can be seen as a component of the Künneth formula isomorphism

Künneth formula generalizations

• The Künneth formula admits various generalizations and extensions to different cohomology theories and algebraic structures
• These generalizations often involve spectral sequences and derived functors, providing a more abstract and powerful framework

Künneth spectral sequence

• The Künneth spectral sequence is a generalization of the Künneth formula that applies to arbitrary cohomology theories
• It is constructed from the derived tensor product of the cohomology theories and converges to the cohomology of the product space
• The Künneth formula can be recovered from the $E_2$ page of the spectral sequence under certain conditions

Künneth formula in homology

• The Künneth formula also holds for homology, relating the homology of a product space to the homology of its factors
• The homological Künneth formula takes a similar form to the cohomological one, with tensor products and torsion products
• In homology, the formula is often easier to apply since homology groups are generally simpler than cohomology groups

Künneth formula for sheaf cohomology

• The Künneth formula can be extended to the setting of sheaf cohomology, which is a cohomology theory for sheaves on topological spaces or algebraic varieties
• In this context, the formula relates the sheaf cohomology of a product of spaces to the sheaf cohomology of the factors
• The sheaf-theoretic Künneth formula involves derived tensor products of sheaves and spectral sequences

Computational examples

• To illustrate the usefulness of the Künneth formula, it is helpful to consider some concrete computational examples
• These examples demonstrate how the formula simplifies the calculation of cohomology groups for product spaces

Torus cohomology via Künneth formula

• Consider the torus $T = S^1 \times S^1$, which is the product of two circles
• The cohomology of the circle is known: $H^0(S^1) = H^1(S^1) = \mathbb{Z}$ and $H^i(S^1) = 0$ for $i > 1$
• Applying the Künneth formula, we obtain: $H^0(T) = \mathbb{Z}$, $H^1(T) = \mathbb{Z} \oplus \mathbb{Z}$, $H^2(T) = \mathbb{Z}$, and $H^i(T) = 0$ for $i > 2$
• The Künneth formula allows us to easily compute the cohomology of the torus from the cohomology of the circle

Projective space products

• The Künneth formula can be used to compute the cohomology of products of projective spaces
• For example, consider the product $\mathbb{CP}^n \times \mathbb{CP}^m$ of complex projective spaces
• The cohomology of $\mathbb{CP}^n$ is known: $H^{2i}(\mathbb{CP}^n) = \mathbb{Z}$ for $0 \leq i \leq n$ and $H^{odd}(\mathbb{CP}^n) = 0$
• Using the Künneth formula, the cohomology of the product can be expressed as a direct sum of tensor products of the cohomology of the factors

Künneth formula and Poincaré duality

• The Künneth formula interacts nicely with Poincaré duality, which relates the cohomology of a manifold to its homology
• For a product of compact oriented manifolds $M \times N$, Poincaré duality gives an isomorphism: $H^k(M \times N) \cong H_{m+n-k}(M \times N)$, where $m = \dim M$ and $n = \dim N$
• Combining this with the Künneth formula, one can express the homology of the product in terms of the homology of the factors
• This combination of Künneth formula and Poincaré duality is a powerful tool in the study of manifolds and their topological invariants