Slant products of cochains with chains #
Let K, L and T be chain complexes of modules over a commutative ring k, indexed by ℕ,
and let D : K ⊗ L ⟶ T be a chain map, for instance the shuffle map
C(X) ⊗ C(Y) ⟶ C(X × Y) of singular chains. A chain z ∈ K_p and a cochain φ : T_n ⟶ P,
for p + q = n, have the slant product φ / z : L_q ⟶ P, the cochain
(φ / z)(v) = φ (D (z ⊗ v)).
Since D is a chain map and the tensor product carries the Koszul signs, the coboundary of the
slant product by a chain z of degree p is δ(φ / z) = (-1)^p (δφ / z - φ / ∂z)
(TauCeti.ChainComplex.d_comp_slantCochain_zero, TauCeti.ChainComplex.slantCochain_d_comp_succ
and TauCeti.ChainComplex.slantCochain_d_zero, which keep track of the degrees in ℕ).
Hence a cocycle slanted by a cycle is a cocycle, a coboundary slanted by a cycle is a coboundary,
and a cocycle slanted by a boundary is a coboundary, so the slant product descends to a
k-bilinear map
Hₚ(K) × Hⁿ(Hom(T, P)) ⟶ H^q(Hom(L, P))
(TauCeti.ChainComplex.slant). It satisfies the evaluation law
⟨φ / x, y⟩ = ⟨φ, D_*(x × y)⟩ for homology classes x ∈ Hₚ(K) and y ∈ H_q(L)
(TauCeti.ChainComplex.kronecker_slant), where x × y is the homology cross product
HomologicalComplex.homologyCross, and it is natural along maps of the triples (K, L, T)
compatible with the chain maps D (TauCeti.ChainComplex.slant_naturality).
Slanting by a cycle z of degree zero is a pullback: if v ↦ e (D (z ⊗ v)) is a chain map
c : L ⟶ T' for some chain map e : T ⟶ T', then e^* φ / [z] = c^* φ
(TauCeti.ChainComplex.slant_homologyπ_zero_homologyMap).
The slant product of singular cohomology, Hⁿ(X × Y) × Hₚ(X) ⟶ H^q(Y), is the case where D is
the shuffle map; it produces, for instance, the classes c₁(U) / γ ∈ H¹(Y) of a line bundle U
on X × Y and loops γ in X.
For D the identity of K ⊗ L, the slant product computes cup products of cochains on the two
factors: (a ⌣ b) / x is b with its coefficients changed along n ↦ μ (⟨a, x⟩ ⊗ n)
(TauCeti.ChainComplex.slant_cup), and it vanishes when the degrees of a and x differ
(TauCeti.ChainComplex.slant_cup_of_ne). Conversely, when the terms and the homology of K are
projective, a class of Hⁿ(Hom(K ⊗ L, P)) is determined by its slant products by the homology
classes of K (TauCeti.ChainComplex.eq_zero_of_forall_slant_eq_zero). Indeed K is then
homotopy equivalent to a complex K' of projectives with zero differentials
(HomologicalComplex.exists_homotopyEquiv_d_eq_zero). A cocycle c of Hom(K' ⊗ L, P) is a
family of maps K'_p ⟶ Hom(L_q, P) into the cocycles of L; when its slant classes vanish these
maps land in the coboundaries, and lift, K'_p being projective, to maps into Hom(L_{q-1}, P)
which assemble into a cochain with coboundary c. With no hypothesis on L, these two facts are
the algebraic content of the Künneth theorem for the cohomology of a product one of whose factors
has finitely generated projective homology.
Main definitions and results #
TauCeti.ChainComplex.slantCochain: the slant product of a chain and a cochain.TauCeti.ChainComplex.d_comp_slantCochain_zero,TauCeti.ChainComplex.slantCochain_d_comp_succandTauCeti.ChainComplex.slantCochain_d_zero: the boundary formulas.TauCeti.ChainComplex.slantCycles: the slant product of a cycle and a cocycle.TauCeti.ChainComplex.slant: the slant product on homology and cohomology, withTauCeti.ChainComplex.slant_homologyπcomputing it on classes of cycles and cocycles.TauCeti.ChainComplex.kronecker_slant: the evaluation law⟨φ / x, y⟩ = ⟨φ, D_*(x × y)⟩.TauCeti.ChainComplex.slant_naturality: naturality;TauCeti.ChainComplex.slant_comp: slanting along a composite.TauCeti.ChainComplex.slant_linearYonedaObjMap: naturality in the coefficients.TauCeti.ChainComplex.slant_homologyπ_zero_homologyMap: the slant product by a cycle of degree zero along whichDrestricts to a chain map.TauCeti.ChainComplex.slantCochain_cupCochain,TauCeti.ChainComplex.slant_cupandTauCeti.ChainComplex.slant_cup_of_ne: slant products of cup products along the identity ofK ⊗ L.TauCeti.ChainComplex.eq_zero_of_forall_slant_eq_zero: slant products by homology classes of a complex with projective terms and homology detect the cohomology ofHom(K ⊗ L, P).
References #
- E. H. Spanier, Algebraic Topology, McGraw–Hill, 1966, Chapter 6 (the slant product).
- 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 of a chain and a cochain along D : K ⊗ L ⟶ T: for p + q = n, the
k-bilinear map sending a chain z ∈ K_p and a cochain φ : T_n ⟶ P to the cochain
v ↦ φ (D (z ⊗ v)) on L_q (TauCeti.ChainComplex.slantCochain_apply).
Instances For
The slant product φ / z evaluates φ on D (z ⊗ v).
The boundary formula for the slant product by a chain of degree zero: the coboundary of
φ / z is δφ / z.
The boundary formula for the slant product by a chain of positive degree: for a chain
w ∈ K_{p+1} and a cochain φ of T, δφ / w = φ / ∂w + (-1)^(p+1) δ(φ / w).
The slant product by a boundary, in degree zero of L: for a chain w ∈ K_{p+1},
φ / ∂w = δφ / w on L_0.
Naturality of the slant product of chains and cochains: if chain maps f : K' ⟶ K,
g : L' ⟶ L and e : T' ⟶ T satisfy (f ⊗ g) ≫ D = D' ≫ e, then slanting the pulled-back
cochain by z along D' is pulling back along g the slant by f z along D.
The slant product of a cycle and a cocycle: the slant product
TauCeti.ChainComplex.slantCochain of the underlying chain and cochain, which is a cocycle by the
boundary formulas (TauCeti.ChainComplex.iCycles_slantCycles).
Instances For
On underlying chains and cochains, the slant product of a cycle and a cocycle is the slant product of the chain and the cochain.
The slant product on homology and cohomology, Hₚ(K) × Hⁿ(Hom(T, P)) ⟶ H^q(Hom(L, P))
for p + q = n, along the chain map D : K ⊗ L ⟶ T: the class of φ / z on the classes of a
cycle z and a cocycle φ (TauCeti.ChainComplex.slant_homologyπ).
Instances For
The slant product on classes: the slant product of the classes of a cycle and a cocycle is the class of their slant product.
The evaluation law of the slant product, ⟨φ / x, y⟩ = ⟨φ, D_*(x × y)⟩: the Kronecker
pairing of the slant product φ / x with a class y ∈ H_q(L) is the Kronecker pairing of φ
with the image under D of the homology cross product x × y ∈ Hₙ(K ⊗ L).
The slant product by a cycle of degree zero: let z be a cycle of degree zero, e : T ⟶ T'
a chain map and c : L ⟶ T' a chain map with c (v) = e (D (z ⊗ v)) in degree q. Then
e^* φ / [z] = c^* φ for every class φ ∈ H^q(Hom(T', P)). For the shuffle map of singular chains,
z a point of X and e induced by the projection X × Y ⟶ Y, the composite
v ↦ e (D (z ⊗ v)) is the identity, so slanting a class pulled back from Y by a point returns it
(TopCat.singularSlant_homologyπ_singularCohomologyMap_snd).
Naturality of the slant product: if chain maps f : K' ⟶ K, g : L' ⟶ L and
e : T' ⟶ T satisfy (f ⊗ g) ≫ D = D' ≫ e, then e^* φ / x = g^* (φ / f_* x) for classes
x ∈ Hₚ(K') and φ ∈ Hⁿ(Hom(T, P)).
The slant product along a composite: for a chain map e : T ⟶ T', slanting along
D ≫ e is slanting along D after pulling the cochain back along e.
The slant product of chains and cochains commutes with changing the coefficients of the
cochain along g : P ⟶ P'.
Naturality of the slant product in the coefficients: for g : P ⟶ P',
g_* φ / x = g_* (φ / x).
The slant product of a cup product of cochains, along the identity of K ⊗ L: slanting
φ ⌣ ψ by z ∈ K_p gives ψ followed by the coefficient map n ↦ μ (φ z ⊗ n).
The slant product of a cup product of cochains of degrees p and q by a chain of degree
p' ≠ p vanishes.
The slant product of a cup product of classes, along the identity of K ⊗ L:
(a ⌣ b) / x is the class b with its coefficients changed along n ↦ μ (⟨a, x⟩ ⊗ n). For
coefficients in the ring itself and μ the multiplication, this is ⟨a, x⟩ • b.
The slant product of a cup product of classes of degrees p and q by a homology class of
degree p' ≠ p vanishes.
Slant products detect the cohomology of a tensor product: if the terms and the homology
objects of K are projective, a class φ of Hⁿ(Hom(K ⊗ L, P)) whose slant products
φ / x ∈ H^q(Hom(L, P)) by all homology classes x ∈ Hₚ(K), p + q = n, vanish is zero. There
is no hypothesis on L.