The Stiefel–Whitney classes of a diagonal form in degrees one and two #
Let K be a field in which 2 is invertible, and write (a) ∈ H¹(G_K, 𝔽₂) for the Kummer class
of a unit a (TauCeti.kummerClass). The Stiefel–Whitney classes of a diagonal quadratic form
⟨a₁, …, aₙ⟩ in degrees one and two are
w₁⟨a₁, …, aₙ⟩ = ∑ᵢ (aᵢ) ∈ H¹(G_K, 𝔽₂), w₂⟨a₁, …, aₙ⟩ = ∑_{i<j} (aᵢ) ∪ (aⱼ) ∈ H²(G_K, 𝔽₂),
the degree-one and degree-two parts of Delzant's total class ∏ᵢ (1 + (aᵢ)). The cup product is
the mod-two Kummer cup pairing TauCeti.kummerCup on square classes, so w₂ is visibly a function
of the square classes of the coefficients.
This file defines both classes on tuples of units, computes them in low rank and on squares, and
expands them on a concatenation of tuples, that is on an orthogonal sum of diagonal forms.
Both classes are unchanged by permuting the coefficients, w₂ because the cup pairing is symmetric
(TauCeti.kummerCup_comm). The class w₁ is the Kummer class of the plain discriminant
a₁ ⋯ aₙ, so it is an invariant of the isometry class of the diagonal form. The invariance of w₂
under isometry, which needs the cup-norm theorem, and the descent of both classes to isometry
classes of regular forms are in TauCeti.FieldTheory.QuadraticForm.StiefelWhitney.Class.
Main definitions #
TauCeti.sw1: the first Stiefel–Whitney class∑ᵢ (aᵢ)of a tuple of units.TauCeti.sw2: the second Stiefel–Whitney class∑_{i<j} (aᵢ) ∪ (aⱼ)of a tuple of units.
Main results #
TauCeti.sw1_eq_kummerClass_prod:w₁⟨a₁, …, aₙ⟩ = (a₁ ⋯ aₙ), the Kummer class of the (unsigned) discriminant.TauCeti.sw1_eq_of_equivalent: isometric diagonal forms have the samew₁.TauCeti.sw1_comp_perm,TauCeti.sw2_comp_perm,TauCeti.PermutationStep.sw1_eqandTauCeti.PermutationStep.sw2_eq: both classes are invariant under permuting the coefficients.TauCeti.BinaryStep.sw1_eq:w₁is invariant under a binary step of a diagonal chain.TauCeti.sw1_fin_zero,TauCeti.sw1_fin_one,TauCeti.sw2_eq_zero_of_le_one,TauCeti.sw1_fin_twoandTauCeti.sw2_fin_two: both classes vanish in rank0,w₁⟨a⟩ = (a),w₂vanishes in rank1,w₁⟨a, b⟩ = (a) + (b)andw₂⟨a, b⟩ = (a) ∪ (b).TauCeti.sw1_appendandTauCeti.sw2_append: on an orthogonal sum,w₁is additive andw₂(q ⊥ r) = w₂(q) + w₂(r) + w₁(q) ∪ w₁(r).TauCeti.sw1_eq_zero_of_isSquareandTauCeti.sw2_eq_zero_of_isSquare: both classes vanish on a tuple of squares, in particular on⟨1, …, 1⟩.
References #
- J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), §3.
- A. Delzant, Définition des classes de Stiefel-Whitney d'un module quadratique sur un corps de caractéristique différente de 2, C. R. Acad. Sci. Paris 255 (1962), 1366–1368.
The first Stiefel–Whitney class of the diagonal form ⟨a₁, …, aₙ⟩: the sum
w₁ = ∑ᵢ (aᵢ) of the Kummer classes of its coefficients, in H¹(G_K, 𝔽₂).
Instances For
The second Stiefel–Whitney class of the diagonal form ⟨a₁, …, aₙ⟩: the sum
w₂ = ∑_{i<j} (aᵢ) ∪ (aⱼ) of the cup products of the Kummer classes of pairs of distinct
coefficients, in H²(G_K, 𝔽₂).
Instances For
Unfolding rule for TauCeti.sw1.
Unfolding rule for TauCeti.sw2.
The first class is the Kummer class of the discriminant #
w₁ read through the Kummer isomorphism on square classes: it is the image of the square
class of the discriminant a₁ ⋯ aₙ.
w₁ is the Kummer class of the discriminant: w₁⟨a₁, …, aₙ⟩ = (a₁ ⋯ aₙ), with the plain
discriminant a₁ ⋯ aₙ rather than the signed one.
w₁ is an isometry invariant: isometric diagonal forms ⟨a₁, …, aₙ⟩ ≅ ⟨b₁, …, bₘ⟩ have
the same first Stiefel–Whitney class, because their discriminants agree modulo squares.
Invariance under permuting the coefficients #
w₁ does not depend on the order of the coefficients.
w₂ does not depend on the order of the coefficients, because the cup pairing is
symmetric.
A permutation step of a diagonal chain does not change w₁.
A permutation step of a diagonal chain does not change w₂.
A binary step of a diagonal chain does not change w₁, since its endpoints are isometric.
Low rank #
The first Stiefel–Whitney class of the rank-zero form vanishes.
The second Stiefel–Whitney class of the rank-zero form vanishes.
w₁⟨a⟩ = (a).
w₂⟨a⟩ = 0.
w₁⟨a, b⟩ = (a) + (b).
w₂⟨a, b⟩ = (a) ∪ (b): the second Stiefel–Whitney class of a binary form is the cup
product of the Kummer classes of its two coefficients.
Orthogonal sums #
The orthogonal-sum formula for w₂: w₂(q ⊥ r) = w₂(q) + w₂(r) + w₁(q) ∪ w₁(r) for the
diagonal forms q = ⟨a₁, …, aₘ⟩ and r = ⟨b₁, …, bₙ⟩. The cross term collects the cups
(aᵢ) ∪ (bⱼ) of the pairs with one coefficient in each form.
Tuples of squares #
w₁⟨1, …, 1⟩ = 0.
w₂⟨1, …, 1⟩ = 0.