The is a key tool in cohomology theory, connecting the cohomology of a product space to its factors. It uses tensor products and 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∙ and D∙, their C∙⊗D∙ is a new cochain complex
The differential on the tensor product is defined using the Leibniz rule: d(c⊗d)=dc⊗d+(−1)degcc⊗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→⨁p+q=nHp(C∙)⊗Hq(D∙)→Hn(C∙⊗D∙)→⨁p+q=n+1Tor(Hp(C∙),Hq(D∙))→0
The direct sum runs over all pairs of degrees p and q such that p+q=n
The 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∙ and D∙
The spectral sequence converges to the cohomology of the total complex, which is isomorphic to H∗(C∙⊗D∙)
By analyzing the E2 page of the spectral sequence, one obtains the Künneth formula
Künneth formula for topological spaces
When applied to the singular cochain complexes of two X and Y, the Künneth formula relates the cohomology of the product space X×Y to the cohomology of X and Y
In this context, the Künneth formula takes the form:
Hn(X×Y)≅⨁p+q=nHp(X)⊗Hq(Y)⊕⨁p+q=n+1Tor(Hp(X),Hq(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=S1×S1 can be computed using the Künneth formula and the known cohomology of the circle S1
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 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 Hp(X)⊗Hq(Y)→Hp+q(X×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 E2 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=S1×S1, which is the product of two circles
The cohomology of the circle is known: H0(S1)=H1(S1)=Z and Hi(S1)=0 for i>1
Applying the Künneth formula, we obtain:
H0(T)=Z, H1(T)=Z⊕Z, H2(T)=Z, and Hi(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 CPn×CPm of complex projective spaces
The cohomology of CPn is known: H2i(CPn)=Z for 0≤i≤n and Hodd(CPn)=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×N, Poincaré duality gives an isomorphism:
Hk(M×N)≅Hm+n−k(M×N), where m=dimM and n=dimN
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