The bidegree-(p, q) component of the cohomology cross product, viewed as
a linear map from H^p(X; R) ⊗ H^q(Y; R) into the total cohomology
⨁_n H^n(X × Y; R) by landing in the degree p + q summand.
Instances For
The total cohomology cross product
× : H^*(X; R) ⊗_R H^*(Y; R) → H^*(X × Y; R), built by distributing the tensor
product over the direct-sum decompositions and assembling the bidegree
components componentCrossMap.
Instances For
Auxiliary statement of the bijectivity of the Kronecker pairing
H^n(X; R) → Hom_R(H_n(X; R), R) when R is a PID and every singular homology
module of X is finitely generated and free. This is the universal coefficient
isomorphism in this special case.
Public version of kroneckerPairing_bijective_of_free_aux: when R is a
PID and H_*(X; R) is finitely generated free in every degree, the Kronecker
pairing H^n(X; R) → Hom_R(H_n(X; R), R) is bijective.
Packaging of kroneckerPairing_bijective_of_free as an explicit linear
equivalence H^n(X; R) ≃ₗ[R] Hom_R(H_n(X; R), R).
Instances For
Total dual of singular homology: the direct sum
⨁_{n : ℕ} Hom_R(H_n(X; R), R). Under the universal coefficient theorem this
is naturally isomorphic to totalSingularCohomology whenever H_*(X; R) is
free.
Instances For
The bidegree-(p, q) component of an inverse to the homology Künneth map
in degree n = p + q: pick a splitting of the Künneth short exact sequence on
X and Y and project the resulting retraction onto the
H_p(X; R) ⊗ H_q(Y; R) summand.
Instances For
Bidegree-(p, q) component of the left edge of the commuting diagram used
to prove bijectivity of the cohomology cross product: apply the Kronecker
pairings on each factor, distribute the tensor product through the dual, and
dualize the Künneth inverse component.
Instances For
The total left edge of the comparison diagram: a linear map
H^*(X; R) ⊗_R H^*(Y; R) → ⨁_n Hom_R(H_n(X × Y; R), R) assembled from the
bidegree components componentDiagramLeftMap.
Instances For
The total right edge of the comparison diagram: apply the Kronecker
pairing in each degree to convert H^*(X × Y; R) into the direct sum of duals
⨁_n Hom_R(H_n(X × Y; R), R).
Instances For
Pointwise commutativity of the comparison diagram on simple tensors and a
single homology class: the Kronecker pairing of the cohomology cross product
a × b against a class z ∈ H_{p+q}(X × Y; R) agrees with the dual pairing
obtained by routing z through the Künneth inverse.
Bidegree-(p, q) commutativity of the comparison diagram: composing the
Kronecker pairing with the cohomology cross product agrees with
componentDiagramLeftMap on H^p(X; R) ⊗ H^q(Y; R).
The total comparison square commutes:
diagramRightMap ∘ totalCohomologyCrossProduct = diagramLeftMap. This is the
totalized version of diagram_comm_component.
The assembly map underlying diagramLeftMap (the direct-sum gluing of all
componentDiagramLeftMap) is bijective; this is the technical core of the
Künneth bijectivity argument on the left edge.
The left edge diagramLeftMap is bijective: combining the bijectivity of
the assembly map with the tensor-direct-sum distributivity equivalence.
Generic helper: if f i : M i →ₗ[R'] N i is bijective for every index i,
then the assembled map ⨁ M i → ⨁ N i built by composing each f i with the
direct-sum inclusion lof is bijective.
The tensor-product term ⨁_{p+q=m} H_p(X) ⊗_R H_q(Y) appearing in the
homology Künneth short exact sequence is free over the PID R whenever
H_*(X; R) is finitely generated and free in every degree.
The Tor term ⨁_{p+q=m-1} Tor^R_1(H_p(X), H_q(Y)) in the homology Künneth
short exact sequence is free over the PID R under the same hypotheses; in
fact, when H_*(X; R) is free this Tor vanishes.
If H_*(X; R) is finitely generated free over a PID R, then
H_m(X × Y; R) is also free, obtained as a direct sum of the (free) Künneth
tensor and Tor terms via a splitting of the Künneth short exact sequence.
Finiteness analogue of kunnethTensorTerm_free: the tensor term in the
homology Künneth sequence is finitely generated over R when each
H_n(X; R) is.
Finiteness analogue of kunnethTorTerm_free: the Tor term in the
homology Künneth sequence is finitely generated over R.
Companion of productHomology_free: H_m(X × Y; R) is finitely generated
over R whenever each H_n(X; R) is, using a splitting of the Künneth short
exact sequence.
The right edge diagramRightMap is bijective: under the freeness and
finiteness assumptions on H_*(X; R), the product homology
H_*(X × Y; R) is also free and finitely generated, so the Kronecker pairing
on X × Y is a bijection in every degree, and bijectivity assembles to the
direct sum.
Diagram-chase proof of the cohomology Künneth theorem: combining the
commutativity diagram_comm with the bijectivity of the left and right edges
forces the total cohomology cross product to be bijective.
The assembly map ⨁_{(p, q)} (H^p(X; R) ⊗ H^q(Y; R)) → H^*(X × Y; R) built
from the bidegree cross-product components is bijective. This is extracted
from the bijectivity of the total cross product by cancelling the
tensor-direct-sum distributivity equivalence.
Cohomology Künneth theorem (Theorem 33.3). Let R be a PID and
suppose every singular homology module H_n(X; R) is finitely generated and
free. Then the cohomology cross product
× : H^*(X; R) ⊗_R H^*(Y; R) → H^*(X × Y; R) is a bijection.