The Künneth theorem for singular cohomology #
For topological spaces X and Y, the cohomology cross products
Hᵖ(X; R, M) ⊗ H^q(Y; S, N) ⟶ Hⁿ(X × Y; R ⊗ S, P) with p + q = n, along a coefficient pairing
μ : M ⊗ N ⟶ P (TauCeti.singularCross), assemble into the cohomology Künneth map
⨁_{p + q = n} Hᵖ(X; R, M) ⊗ H^q(Y; S, N) ⟶ Hⁿ(X × Y; R ⊗ S, P)
(TopCat.singularCohomologyKunneth), whose source is the degree n part of the tensor product of
the graded k-modules H*(X; R, M) and H*(Y; S, N).
With coefficients in a commutative ring k itself, suppose that the homology modules Hₚ(X; k)
are projective, and finitely generated for p ≤ n. Then the Kronecker map
Hᵖ(X; k) ⟶ Hom(Hₚ(X; k), k) is an isomorphism
(TopCat.singularKroneckerEquivOfProjectiveHomology), and so is the canonical map
Hom(Hₚ(X; k), k) ⊗ H^q(Y; k) ⟶ Hom(Hₚ(X; k), H^q(Y; k)) (dualTensorHomEquiv). The slant
products by the classes of Hₚ(X; k) (TopCat.singularSlant) send a class of
Hⁿ(X × Y; k) to ⨁_{p + q = n} Hom(Hₚ(X; k), H^q(Y; k)). On cross products they are computed
by the Kronecker pairing, (a × b) / x = ⟨a, x⟩ • b, and vanish when the degrees of a and x
differ (TopCat.singularSlant_singularCross, TopCat.singularSlant_singularCross_of_ne); so the
slant products split the Künneth map. They are moreover injective on Hⁿ(X × Y; k)
(TopCat.eq_zero_of_forall_singularSlant_eq_zero). This gives the Künneth theorem for
cohomology (TopCat.singularCohomologyKunnethIso): the cross product induces an isomorphism
⨁_{p + q = n} Hᵖ(X; k) ⊗ H^q(Y; k) ≅ Hⁿ(X × Y; k).
No hypothesis is placed on Y, and no Tor term appears: over ℤ this covers every X with
free homology of finite rank, for instance spheres
(TopCat.singularCohomologyKunneth_sphere_bijective, in
TauCeti.AlgebraicTopology.Cohomology.Sphere.Basic), but not, say, real projective space. Over
a commutative semisimple ring, for instance a field, every module is projective
(Module.projective_of_isSemisimpleRing), so there only the finite generation of Hₚ(X; k) for
p ≤ n is needed, for instance when X has finite CW type
(TauCeti.finite_singularHomology_of_finiteCWType). Here Hⁿ(X × Y; k) is computed from the
singular chains C(X × Y; k ⊗ k), whose coefficient module k ⊗ k ≅ k is the one carried by the
cross product, and the pairing k ⊗ k ⟶ k is multiplication.
Main definitions and results #
TopCat.singularCohomologyKunneth: the cohomology Künneth map, characterized on summands byTopCat.singularCohomologyKunneth_ιTensorObj_tmul, withTopCat.singularCohomologyKunneth_naturalityits naturality in both spaces.TopCat.singularCohomologyKunnethIso,TopCat.isIso_singularCohomologyKunnethandTopCat.singularCohomologyKunneth_bijective: the Künneth theorem for cohomology over a commutative ring, when the homology of one factor is projective and finitely generated.
References #
- A. Hatcher, Algebraic Topology, Section 3.2, the cross product and Theorem 3.15, and Section 3.B, the Künneth formulas.
- E. H. Spanier, Algebraic Topology, McGraw–Hill, 1966, Chapter 6 (slant products).
The cohomology Künneth map ⨁_{p + q = n} Hᵖ(X; R, M) ⊗ H^q(Y; S, N) ⟶ Hⁿ(X × Y; R ⊗ S, P),
whose restriction to the summand Hᵖ(X; R, M) ⊗ H^q(Y; S, N) is the cohomology cross product
TauCeti.singularCross along the coefficient pairing μ.
Instances For
The cohomology Künneth map restricts to the cohomology cross product on each summand.
Naturality of the cohomology Künneth map in both spaces: for continuous maps
f : X ⟶ X' and g : Y ⟶ Y', pulling back by f^* ⊗ g^* and then applying the Künneth map of
X and Y is applying the Künneth map of X' and Y' and then pulling back by (f × g)^*.
Naturality of the cohomology Künneth map in both spaces: for continuous maps
f : X ⟶ X' and g : Y ⟶ Y', pulling back by f^* ⊗ g^* and then applying the Künneth map of
X and Y is applying the Künneth map of X' and Y' and then pulling back by (f × g)^*.
The Künneth isomorphism for cohomology: over a commutative ring k, if the homology
Hₚ(X; k) is projective in every degree and finitely generated in degrees p ≤ n, the cohomology
cross product induces an isomorphism ⨁_{p + q = n} Hᵖ(X; k) ⊗ H^q(Y; k) ≅ Hⁿ(X × Y; k), for
every space Y. Here the product is computed from singular chains with coefficients in
k ⊗ k ≅ k, and the pairing k ⊗ k ⟶ k is multiplication. Over a commutative semisimple ring,
for instance a field, every module is projective (Module.projective_of_isSemisimpleRing), so
only the finite generation of Hₚ(X; k) is needed there.
Instances For
Over a commutative ring, the cohomology Künneth map is an isomorphism when the homology of X
is projective, and finitely generated in degrees p ≤ n.
The Künneth theorem for cohomology: over a commutative ring k, if the homology Hₚ(X; k)
is projective in every degree and finitely generated in degrees p ≤ n, for instance if it is
free of finite rank as for spheres and tori, the cohomology Künneth map
⨁_{p + q = n} Hᵖ(X; k) ⊗ H^q(Y; k) ⟶ Hⁿ(X × Y; k) is bijective for every space Y.