The slant product in singular cohomology #
For spaces X and Y, modules R, S, T and M over a commutative ring k and a
coefficient map u : R ⊗ S ⟶ T, the slant product
Hₚ(X; R) × Hⁿ(X × Y; T, M) ⟶ H^q(Y; S, M), p + q = n,
sends a homology class x of X and a cohomology class φ of the product to the class φ / x
on Y represented by the cochain σ ↦ φ (u_*(z × σ)), where z is a cycle representing x and
z × σ is the shuffle product of singular chains. It is the slant product of chains and cochains
TauCeti.ChainComplex.slant along the shuffle map followed by u, with X as the first factor.
The slant product satisfies the evaluation law
⟨φ / x, y⟩ = ⟨φ, u_*(x × y)⟩ (TopCat.singularKronecker_singularSlant), where x × y is the
homology cross product TopCat.singularHomologyCross; on the classes of a cycle and a cocycle it
is the class of the slant product of representatives (TopCat.singularSlant_homologyπ), and it is
natural, (f × g)^* φ / x = g^* (φ / f_* x) (TopCat.singularSlant_naturality).
Slant products by homology classes turn a cohomology class on a product into classes on the second
factor. For a line bundle U on X × Y with first Chern class c₁(U) ∈ H²(X × Y; ℤ), the slant
by the class of a point of X is a class of degree two on Y, and the slant by a loop in X is a
class of degree one on Y; these are the point and loop classes of universal line bundles over
moduli spaces.
On classes pulled back from one factor the slant product is computed by the cohomology of a point:
- a class
pr₁^* apulled back fromXhaspr₁^* a / x = 0wheneverq ≠ 0(TopCat.singularSlant_singularCohomologyMap_fst_eq_zero), because by naturality it is pulled back from the slant product onX × pt, which lies inH^q(pt) = 0; - a class
pr₂^* bpulled back fromYhaspr₂^* b / x = 0wheneverp ≠ 0(TopCat.singularSlant_singularCohomologyMap_snd_eq_zero), because by naturality it is the slant product by the image ofxinHₚ(pt) = 0; - for the class
xof a point ofXwith coefficientr,pr₂^* b / xisbwith its coefficients changed alongs ↦ u (r ⊗ s)(TopCat.singularSlant_homologyπ_singularCohomologyMap_snd), and it isbitself whenu (r ⊗ s) = s(TopCat.singularSlant_homologyπ_singularCohomologyMap_snd_eq_self). This uses the shuffle product of a vertex with a simplex (TopCat.ιChainComplex_tensorHom_ιChainComplex_shuffle_f_snd).
So if a class on X × Y is pr₁^* e + φ, for instance the first Chern class
c₁(pr₁^* E ⊗ U) = pr₁^* c₁(E) + c₁(U) of a line bundle twisted by a line bundle E on X, then
its slant products by the point class (p = 0, q = 2) and by loops (p = 1, q = 1) are those
of φ: the class pr₁^* e contributes to neither. Classes pulled back from Y survive the slant
by a point and are killed by the slant by a loop.
Slant products of cohomology cross products are computed by the Kronecker pairing: for
a ∈ Hᵖ(X; R, M), b ∈ H^q(Y; S, N) and a pairing μ : M ⊗ N ⟶ P, the slant of a × b by
x ∈ Hₚ(X; R) is b with its coefficients changed along n ↦ μ (⟨a, x⟩ ⊗ n)
(TopCat.singularSlant_singularCross; for a with coefficients in k and the scalar
multiplication k ⊗ N ⟶ N as pairing, (a × b) / x = ⟨a, x⟩ • b,
TopCat.singularSlant_singularCross_leftUnitor), and the slant by a
class of another degree vanishes
(TopCat.singularSlant_singularCross_of_ne). Conversely, when R and the homology modules
Hₚ(X; R) are projective, a class of Hⁿ(X × Y; R ⊗ S, M) is determined by its slant products
by the homology classes of X (TopCat.eq_zero_of_forall_singularSlant_eq_zero), whatever the
space Y. Through Eilenberg–Zilber both statements are those of the tensor product of singular
chain complexes, TauCeti.ChainComplex.slant_cup and
TauCeti.ChainComplex.eq_zero_of_forall_slant_eq_zero. Together they give the Künneth theorem
for the cohomology of a product one of whose factors has finitely generated projective homology,
TopCat.singularCohomologyKunnethIso.
Along the multiplication k ⊗ k ≅ k these results characterize the classes of Hⁿ(X × Y; M)
pulled back from Y, for X path-connected with projective homology and x₀ ∈ X: slanting by
the class [x₀] of the point returns a pulled-back class,
pr₂^* b / [x₀] = b (TopCat.singularSlant_singularHomology₀Section_singularCohomologyMap_snd),
and a class φ whose slant products by all classes of positive degree vanish is pr₂^* (φ / [x₀])
(TopCat.singularCohomologyMap_snd_singularSlant_of_forall_ne_zero). The slant products of a
cup product pr₁^* a ⌣ pr₂^* c are those of the cross product a × c
(TopCat.singularSlant_singularCup_singularCohomologyMap_fst_snd).
Main definitions and results #
TopCat.singularSlant: the slant productHₚ(X; R) × Hⁿ(X × Y; T, M) ⟶ H^q(Y; S, M).TopCat.singularSlant_homologyπ: its value on classes of cycles and cocycles.TopCat.singularKronecker_singularSlant: the evaluation law⟨φ / x, y⟩ = ⟨φ, u_*(x × y)⟩.TopCat.singularSlant_naturality: naturality in both spaces.TopCat.singularSlant_coefficient_naturality: slanting a class with changed coefficientsf^* φalonguis slantingφalongu ≫ f.TopCat.singularSlant_linearYonedaObjMap: naturality in the coefficient moduleM.TopCat.singularSlant_singularCohomologyMap_fst_eq_zeroandTopCat.singularSlant_singularCohomologyMap_snd_eq_zero: the vanishing of the slant product of a class pulled back from one factor in the degrees where the cohomology or homology of a point vanishes.TopCat.singularSlant_homologyπ_singularCohomologyMap_sndandTopCat.singularSlant_homologyπ_singularCohomologyMap_snd_eq_self: the slant product of a class pulled back fromYby the class of a point ofX, restated for the point classesTauCeti.singularHomology₀SectioninTopCat.singularSlant_singularHomology₀Section_singularCohomologyMap_snd_eq_self.TopCat.singularSlant_eq_slant: through Eilenberg–Zilber, the slant product is the slant product on the tensor product of singular chain complexes.TopCat.singularSlant_singularCross,TopCat.singularSlant_singularCross_of_neandTopCat.singularSlant_singularCross_leftUnitor: slant products of cross products.TopCat.eq_zero_of_forall_singularSlant_eq_zero: slant products by the homology of a factor with projective homology detect the cohomology of the product;TopCat.eq_zero_of_forall_singularSlant_eq_zero_of_isIsois the same along an isomorphismu : R ⊗ S ≅ T, for instancek ⊗ k ≅ k.TopCat.mem_range_singularCohomologyMap_snd_iff: forXpath-connected with projective homology, a class ofHⁿ(X × Y; M)is pulled back fromYif and only if its slant products by all classes of positive degree vanish.TopCat.singularSlant_singularHomology₀Section_singularCohomologyMap_snd_add_singularCupandTopCat.singularSlant_singularCohomologyMap_snd_add_singularCup: the slant products ofpr₂^* b + pr₁^* a ⌣ pr₂^* cby a point and in the degree ofa.
References #
- E. H. Spanier, Algebraic Topology, McGraw–Hill, 1966, Chapter 6 (the slant product of singular homology and cohomology).
- A. Dold, Lectures on Algebraic Topology, Springer, 1972, Chapter VII (products).
- A. Hatcher, Algebraic Topology, Theorem 3.15 and Section 3.B, the Künneth formulas.
The slant product Hₚ(X; R) × Hⁿ(X × Y; T, M) ⟶ H^q(Y; S, M) for p + q = n, along the
coefficient map u : R ⊗ S ⟶ T: the slant product of chains and cochains along the shuffle map
C(X; R) ⊗ C(Y; S) ⟶ C(X × Y; R ⊗ S) followed by the map induced by u.
Instances For
The slant product of singular cohomology is the slant product of chains and cochains along the shuffle map followed by the coefficient map.
The slant product on classes: the slant product of the classes of a singular cycle z and
a singular cocycle φ is the class of the cochain σ ↦ φ (u_*(z × σ)).
The evaluation law of the slant product, ⟨φ / x, y⟩ = ⟨φ, u_*(x × y)⟩: the Kronecker
pairing of the slant product φ / x ∈ H^q(Y; S, M) with a class y ∈ H_q(Y; S) is the
Kronecker pairing of φ with the image under u of the homology cross product
x × y ∈ Hₙ(X × Y; R ⊗ S).
Naturality of the slant product in both spaces: for continuous maps f : X ⟶ X' and
g : Y ⟶ Y', (f × g)^* φ / x = g^* (φ / f_* x).
Naturality of the slant product in the coefficients: for coefficient maps
u : R ⊗ S ⟶ T and f : T ⟶ T', slanting along u the class f^* φ, whose cochains are those
of φ precomposed with the chain map induced by f, is slanting φ along u ≫ f.
Naturality of the slant product in the coefficients: for a map of coefficient modules
g : M ⟶ M', g_* φ / x = g_* (φ / x).
A class pulled back from the first factor has no slant products in positive codegree: for
a ∈ Hⁿ(X; T, M) and x ∈ Hₚ(X; R) with p + q = n and q ≠ 0, pr₁^* a / x = 0. In particular
pr₁^* a contributes nothing to the slant products by the class of a point (p = 0) or by a loop
(p = 1) of a class of degree two.
A class pulled back from the second factor has no slant products by classes of positive
degree: for b ∈ Hⁿ(Y; T, M) and x ∈ Hₚ(X; R) with p + q = n and p ≠ 0,
pr₂^* b / x = 0. In particular pr₂^* b has no slant products by loops.
The slant product of a class pulled back from the second factor by the class of a point:
let z be the singular 0-cycle of X given by a singular vertex x with coefficient r ∈ R,
and let c : S ⟶ T be the coefficient map s ↦ u (r ⊗ s). Then for b ∈ H^q(Y; T, M),
pr₂^* b / [z] = c^* b ∈ H^q(Y; S, M), the class b with its coefficients changed along c.
Slanting a class pulled back from the second factor by a point returns the class: for the
singular 0-cycle z of X given by a singular vertex with coefficient r, and a coefficient map
u : R ⊗ S ⟶ S with u (r ⊗ s) = s, such as the unit 1 of a ring R = S = k and the
multiplication k ⊗ k ⟶ k, pr₂^* b / [z] = b for every b ∈ H^q(Y; S, M).
Slanting a class pulled back from the second factor by the class of a point returns the
class: pr₂^* b / [x₀] = b for the class [x₀] ∈ H₀(X; R) of a point x₀ with coefficient r
(TauCeti.singularHomology₀Section), along a coefficient map u : R ⊗ S ⟶ S with
u (r ⊗ s) = s.
The slant product through Eilenberg–Zilber: slanting a class φ of X × Y by x is
slanting its image shuffle^* φ in the cohomology of Hom(C(X; R) ⊗ C(Y; S), M) by x along the
identity.
The slant product of a cross product: for a ∈ Hᵖ(X; R, M), b ∈ H^q(Y; S, N), a pairing
μ : M ⊗ N ⟶ P and x ∈ Hₚ(X; R), the slant (a × b) / x is b with its coefficients changed
along n ↦ μ (⟨a, x⟩ ⊗ n). For coefficients in k and μ the multiplication, (a × b) / x is
⟨a, x⟩ • b.
The slant product of a cross product of classes of degrees p and q by a homology class of
degree p' ≠ p vanishes.
The slant product of a cross product with coefficients in k: for a ∈ Hᵖ(X; k),
b ∈ H^q(Y; N) and x ∈ Hₚ(X; k), with the scalar multiplication k ⊗ N ⟶ N as pairing,
(a × b) / x = ⟨a, x⟩ • b.
Slant products detect the cohomology of a product: if the coefficient module R and the
homology modules Hₚ(X; R) are projective, a class φ ∈ Hⁿ(X × Y; R ⊗ S, M) whose slant products
φ / x ∈ H^q(Y; S, M) by all classes x ∈ Hₚ(X; R), p + q = n, vanish is zero. There is no
hypothesis on Y.
Slant products along an isomorphism detect the cohomology of a product: if R and the
homology modules Hₚ(X; R) are projective and u : R ⊗ S ⟶ T is an isomorphism, a class
φ ∈ Hⁿ(X × Y; T, M) whose slant products along u by all classes of Hₚ(X; R) vanish is zero.
For R = S = T = k and u the multiplication k ⊗ k ≅ k, this applies to the cohomology of
X × Y with coefficients in M, computed from the singular chains with coefficients in k.
Slanting a class pulled back from Y by a point of X returns it:
pr₂^* b / [x] = b, for the class [x] ∈ H₀(X; k) of a point and the slant product along the
multiplication k ⊗ k ≅ k.
A class with no slant products in positive degree is pulled back: let X be
path-connected with projective homology and x₀ ∈ X. If φ ∈ Hⁿ(X × Y; M) has φ / x = 0 for
every class x ∈ Hₚ(X; k) with p ≠ 0, then φ = pr₂^* (φ / [x₀]).
Classes pulled back from the second factor: for X path-connected with projective
homology, a class φ ∈ Hⁿ(X × Y; M) is pulled back from Y if and only if its slant products by
all classes of Hₚ(X; k) with p ≠ 0 vanish.
The slant product of a cup product of pulled-back classes: for a ∈ Hᵖ(X; k) and
b ∈ H^q(Y; M), the slant product of pr₁^* a ⌣ pr₂^* b ∈ Hⁿ(X × Y; M) along k ⊗ k ≅ k is the
slant product of the cross product a × b, which TopCat.singularSlant_singularCross_leftUnitor
and TopCat.singularSlant_singularCross_of_ne compute.
Slanting pr₂^* b + pr₁^* a ⌣ pr₂^* c by a point: for a ∈ Hᵖ(X; k) with p ≠ 0, the
slant product of pr₂^* b + pr₁^* a ⌣ pr₂^* c ∈ Hⁿ(X × Y; M) by the class [x] ∈ H₀(X; k) of a
point is b.
Slanting pr₂^* b + pr₁^* a ⌣ pr₂^* c in the degree of a: for a ∈ Hᵖ(X; k) with
p ≠ 0 and x ∈ Hₚ(X; k), the slant product of pr₂^* b + pr₁^* a ⌣ pr₂^* c ∈ Hⁿ(X × Y; M) by
x is ⟨a, x⟩ • c.