The relator functional of a minimal presentation and the cup product #
Let F = freeProP p X be a free pro-p group, let R ≤ Φ(F) be a closed normal subgroup
contained in its pro-p Frattini subgroup, and let G ≅ F ⧸ R, so that 1 → R → F → G → 1 is a
minimal presentation of G. The transgression H¹(R, 𝔽_p)^F → H²(G, 𝔽_p) is bijective and
H¹(R, 𝔽_p)^F is the continuous 𝔽_p-dual of R ⧸ Rᵖ[R, F] (TauCeti.h2DualEquiv). Evaluating at
a relator r ∈ R therefore defines a linear functional on H²(G, 𝔽_p), the relator functional
TauCeti.freeProP.relatorFunctional; TauCeti.freeProP.relatorEval is the same functional on the
explicit model H2 G (ZMod p), for an arbitrary trivial action of G on 𝔽_p.
The main theorem computes the relator functional on cup products. For a, b ∈ H¹(G, 𝔽_p), with
characters χ, ψ : F → 𝔽_p obtained by composing the characters of G attached to a and b with
F → G,
relatorFunctional r (a ⌣ b) = - B_{⟦r⟧}(χ, ψ),
where B_{⟦r⟧} is the degree-one form (TauCeti.freeProP.degreeOneForm) of the class of r in
gr_1(F) = λ_1(F) ⧸ λ_2(F). This is Labute's Proposition 3: the cup product on H¹(G, 𝔽_p) is read
off the coordinates of the relator class in the standard basis of gr_1(F), the commutator
coordinates off the diagonal and (p choose 2) times the p-power coordinates on it. The proof
lifts the cup product through a Heisenberg cochain h of (χ, ψ), which exists because
H²(F, 𝔽_p) = 0; the transgression formula tg [-h|_R] = a ⌣ b is
TauCeti.ContCohomology.IsHeisenbergCochain.transgression_negRestrict, and on λ_1(F) the cochain
h is the Heisenberg functional of (χ, ψ)
(TauCeti.ContCohomology.IsHeisenbergCochain.apply_eq_heisenbergFunctional).
When R is the closed normal closure of the single relator r, the relator functional is injective
(TauCeti.freeProP.relatorFunctional_injective): R ⧸ Rᵖ[R, F] is then topologically generated by
the class of r, so a character of it vanishing at r vanishes identically. For a one-relator
group with H²(G, 𝔽_p) ≠ 0 this makes the relator functional the trace isomorphism
H²(G, 𝔽_p) ≅ 𝔽_p attached to the relator.
Main definitions #
TauCeti.freeProP.relatorEval: evaluation of the inverse transgression at a relator, on the explicit modelH2 G (ZMod p).TauCeti.freeProP.relatorFunctional: the relator functionalH²(G, 𝔽_p) → 𝔽_p, on the canonical carriercohomFp p G 2.
Main results #
TauCeti.ContCohomology.IsHeisenbergCochain.apply_eq_heisenbergFunctional: a Heisenberg cochain of two𝔽_p-characters of a free pro-pgroup of finite rank agrees onλ_1(F)with their Heisenberg functional.TauCeti.freeProP.relatorEval_h2QuotientEquiv_transgression,TauCeti.freeProP.relatorFunctional_apply: the defining equations.TauCeti.freeProP.relatorEval_explicitCup11,TauCeti.freeProP.relatorFunctional_cupFp: the relator functional of a cup product is minus the degree-one form of the relator class.TauCeti.freeProP.relatorEval_injective,TauCeti.freeProP.relatorFunctional_injective: for a single relator, the relator functional is injective.
References #
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §1.4 and Proposition 3.
- J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.9.5) and Chapter III, §9.
Heisenberg cochains and the Heisenberg functional #
A Heisenberg cochain of two 𝔽_p-characters of a free pro-p group is their Heisenberg
functional on λ_1(F). Let F be the free pro-p group on a finite type, let a, b be
continuous 1-cocycles for a trivial action on 𝔽_p, that is continuous characters χ, ψ of
F, and let h be a Heisenberg cochain for (a, b) and multiplication. Then on n ∈ λ_1(F) the
value h n is the Heisenberg functional of χ and ψ at the class of n in gr_1(F).
The relator functional #
Evaluation at a relator, on the explicit model. Let F be the free pro-p group on X,
let R ≤ Φ(F) be a closed normal subgroup, and let G ≅ F ⧸ R be a group acting trivially on
𝔽_p, as does F. A class in H²(G, 𝔽_p) corresponds under the inverse transgression to a
character of R ⧸ Rᵖ[R, F] (TauCeti.h2DualEquiv); evaluating that character at the class of a
relator r ∈ R is the relator functional of r, an additive map H2 G (ZMod p) →+ ZMod p.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The defining equation of TauCeti.freeProP.relatorEval: the value at the class of r of the
character of R ⧸ Rᵖ[R, F] attached to x by TauCeti.h2DualEquiv.
The relator functional inverts the transgression: on the class transgressed from an
invariant class y ∈ H¹(R, 𝔽_p)^F, the relator functional of r is the value at r of the
continuous homomorphism R → 𝔽_p representing y.
The relator functional of a single relator is injective. If R is the closed normal
closure of r, then R ⧸ Rᵖ[R, F] is topologically generated by the class of r, so a character
of it vanishing at r is trivial.
Labute's Proposition 3, on the explicit model. Let F be the free pro-p group on a
finite type X, let R ≤ Φ(F) be a closed normal subgroup, let G ≅ F ⧸ R act trivially on 𝔽_p,
as does F, and let r ∈ R. For two classes a, b ∈ H¹(G, 𝔽_p), the relator functional of r on
the explicit cup product a ⌣ b is minus the degree-one form of the class of r in gr_1(F),
evaluated at the characters of F obtained from a and b by composing with F → G.
The relator functional. Let F be the free pro-p group on X, let R ≤ Φ(F) be a
closed normal subgroup, and let G ≅ F ⧸ R, so that 1 → R → F → G → 1 is a minimal presentation
of G. A class in H²(G, 𝔽_p) corresponds under the inverse transgression to a continuous
homomorphism R ⧸ Rᵖ[R, F] → 𝔽_p; evaluating it at the class of a relator r ∈ R is the relator
functional of r, an 𝔽_p-linear functional on H²(G, 𝔽_p). On a cup product it is minus the
degree-one form of the class of r in gr_1(F) (TauCeti.freeProP.relatorFunctional_cupFp), and
for a single relator it is injective (TauCeti.freeProP.relatorFunctional_injective).
Equations
- TauCeti.freeProP.relatorFunctional hRc hR e r = AddMonoidHom.toZModLinearMap p ((TauCeti.freeProP.relatorEval hRc hR e ⋯ ⋯ r).comp (TauCeti.cohomFpAddEquivH2 p G ⋯).toAddMonoidHom)
Instances For
The defining equation of the relator functional: on the canonical carrier it is the
explicit relator functional TauCeti.freeProP.relatorEval, for the trivial actions
trivialZModAction, composed with the identification cohomFpAddEquivH2 of H²(G, 𝔽_p) with its
explicit model.
The relator functional of a single relator is injective. If R is the closed normal
closure of r, the relator functional of r is injective on H²(G, 𝔽_p).
Labute's Proposition 3: the relator functional of a cup product is minus the degree-one form
of the relator class. Let F be the free pro-p group on a finite type X, let R ≤ Φ(F) be a
closed normal subgroup, let G ≅ F ⧸ R, and let r ∈ R. For a, b ∈ H¹(G, 𝔽_p), with characters
χ, ψ : F → 𝔽_p obtained by composing the characters of G attached to a and b with F → G,
relatorFunctional r (a ⌣ b) = - B_{⟦r⟧}(χ, ψ),
where B_{⟦r⟧} is the degree-one form of the class of r in gr_1(F): the cup product on
H¹(G, 𝔽_p) is read off the coordinates of the relator class in the standard basis of gr_1(F).