Singular cohomology of spheres and of products with spheres #
For every commutative ring k and every k-module M, evaluation on the native standard
homology generator identifies Hⁿ⁺¹(Sⁿ⁺¹; M) with M. Positive-degree cohomology vanishes
outside the sphere dimension, including for the zero-sphere. No hereditary-ring or
injectivity assumption is needed: the singular homology of a sphere with coefficients in
k is free, so the Kronecker map is an isomorphism over any commutative ring.
The equivalence is natural in the coefficient module. Its evaluation formula uses exactly
TauCeti.reducedSingularHomologyTopCatSphereIso and the canonical reduced-to-ordinary
inclusion; it does not choose another orientation or generator. In particular its inverse
applied to a nonzero coefficient gives a nonzero top cohomology class.
Dually to the standard generator TauCeti.singularHomologyTopCatSphereGenerator of
Hₙ₊₁(Sⁿ⁺¹; k), the class TauCeti.singularCohomologyTopCatSphereGenerator of Hⁿ⁺¹(Sⁿ⁺¹; k)
takes the value 1 on it.
Since the homology of a sphere is free of finite rank, the Künneth theorem for cohomology applies
to a product with a sphere over every commutative ring and with no hypothesis on the other factor
(TopCat.singularCohomologyKunneth_sphere_bijective).
For a sphere of positive dimension m + 1 this decomposition is made explicit, with the summands
named as pullbacks and cup products (TopCat.singularCohomologySphereProdEquiv): for
m + 1 + q = n,
Hⁿ(Y; k) × H^q(Y; k) ≃ Hⁿ(S^{m+1} × Y; k), (b, c) ↦ pr₂^* b + pr₁^* θ ⌣ pr₂^* c,
where θ ∈ H^{m+1}(S^{m+1}; k) evaluates to 1 on the standard generator of the homology of the
sphere. The inverse sends a class to its slant products by the class of a point and by the
standard generator; that these two slant products determine the class is the detection theorem
TopCat.eq_zero_of_forall_singularSlant_eq_zero, since the two classes span the homology of the
sphere. For k = ℤ and the circle this splits H²(S¹ × Y; ℤ) into pr₂^* H²(Y; ℤ) and
θ ⌣ pr₂^* H¹(Y; ℤ), with all torsion of Y retained.
References #
- A. Hatcher, Algebraic Topology, Section 3.1 (universal coefficients) and Section 3.2 (the cohomology of spheres).
Evaluation on the standard homology generator identifies the top cohomology of a positive-dimensional sphere with its coefficient module, over any commutative ring.
Instances For
The sphere cohomology equivalence evaluates on the native reduced generator, included in ordinary homology.
The inverse sphere cohomology equivalence has prescribed evaluation on the native standard generator.
The dual of the sphere generator: the class of Hⁿ⁺¹(TopCat.sphere (n + 1); k) taking
the value 1 on the standard generator TauCeti.singularHomologyTopCatSphereGenerator. It
generates Hⁿ⁺¹(TopCat.sphere (n + 1); k) ≅ k. Both coefficients are written as the monoidal unit
𝟙_ (ModuleCat k), the form in which cup and slant products take them.
Instances For
The dual of the sphere generator pairs to 1 with the standard generator.
Positive-degree singular cohomology of a sphere vanishes away from its dimension, with arbitrary module coefficients over any commutative ring.
Evaluation on the sphere generator commutes with every coefficient-module map. The map on cohomology is induced by postcomposition on the actual singular cochains.
The cohomology of a product with a sphere: over every commutative ring k, the cohomology
cross product induces an isomorphism ⨁_{p + q = n} Hᵖ(Sᵈ; k) ⊗ H^q(Y; k) ≅ Hⁿ(Sᵈ × Y; k) for
every space Y, since the homology of the sphere is free of finite rank. For d = 1 and
k = ℤ this splits Hⁿ(S¹ × Y; ℤ) into the cross products of H⁰(S¹; ℤ) ≅ ℤ with Hⁿ(Y; ℤ) and
of H¹(S¹; ℤ) ≅ ℤ with Hⁿ⁻¹(Y; ℤ), torsion included.
The cohomology of a product with a sphere of positive dimension m + 1, with
coefficients in a commutative ring k: for m + 1 + q = n,
Hⁿ(Y; k) × H^q(Y; k) ≃ Hⁿ(S^{m+1} × Y; k), (b, c) ↦ pr₂^* b + pr₁^* θ ⌣ pr₂^* c,
where θ ∈ H^{m+1}(S^{m+1}; k) is the class dual to the standard generator of the homology of the
sphere (TauCeti.singularCohomologyTopCatSphereEquiv). The inverse sends a class φ to its slant
products (φ / [x₀], φ / [S^{m+1}]) by the class of a point and by the standard generator
(TopCat.singularCohomologySphereProdEquiv_symm_apply). No hypothesis is placed on Y, and
torsion is retained: for k = ℤ, m = 0 and n = 2 this is the splitting
H²(S¹ × Y; ℤ) = pr₂^* H²(Y; ℤ) ⊕ (θ ⌣ pr₂^* H¹(Y; ℤ)).
Instances For
The splitting of the cohomology of S^{m+1} × Y sends (b, c) to
pr₂^* b + pr₁^* θ ⌣ pr₂^* c.
The inverse of the splitting of the cohomology of S^{m+1} × Y sends a class φ to its slant
products by the class of any point x₀ and by the standard generator of H_{m+1}(S^{m+1}; k).