Labute's criterion: the one-relator pro-p groups that are Demushkin

Let F = freeProP p X be the free pro-p group on a finite type X, let r ∈ Φ(F) be a relator in its pro-p Frattini subgroup, and let G ≅ ⟨X ∣ r⟩ = F ⧸ ⟪r⟫ be the one-relator pro-p group it presents, a minimal presentation. The relator functional TauCeti.freeProP.relatorFunctional of r is injective on H²(G, 𝔽_p) and sends the cup product a ⌣ b of two classes of H¹(G, 𝔽_p) to minus the degree-one form TauCeti.freeProP.degreeOneForm of the class ⟦r⟧ ∈ gr_1(F), evaluated at the characters of F attached to a and b. Since every 𝔽_p-character of F kills Φ(F) ⊇ ⟪r⟫, the characters of G are exactly the characters of F, and the cup form of G on H¹(G, 𝔽_p) is, up to sign, the degree-one form of ⟦r⟧ on the continuous dual of F: the cup product a ⌣ b vanishes exactly when B_{⟦r⟧}(χ_a, χ_b) does (TauCeti.freeProP.cupFp_eq_zero_iff_degreeOneForm_eq_zero).

Labute's criterion follows: G is a Demushkin group exactly when X is nonempty and the degree-one form of ⟦r⟧ is nondegenerate (TauCeti.isDemushkin_iff_nondegenerate_degreeOneForm), and the cup form is alternating exactly when the degree-one form is (TauCeti.freeProP.isAlt_degreeOneForm_iff_forall_cupFp_self_eq_zero). The consequences for the relator of a Demushkin group, Labute's normal forms modulo λ_2(F), are drawn in TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.Criterion.

Main results

• TauCeti.freeProP.cupFp_eq_zero_iff_degreeOneForm_eq_zero: the cup product of two classes of H¹(G, 𝔽_p) vanishes exactly when the degree-one form of the relator class vanishes at the attached characters of F.
• TauCeti.freeProP.isAlt_degreeOneForm_iff_forall_cupFp_self_eq_zero: the degree-one form of the relator class is alternating exactly when every cup square a ⌣ a on H¹(G, 𝔽_p) vanishes.
• TauCeti.IsDemushkin.nondegenerate_degreeOneForm, TauCeti.isDemushkin_of_nondegenerate_degreeOneForm, TauCeti.isDemushkin_iff_nondegenerate_degreeOneForm: Labute's criterion, a one-relator pro-p group ⟨X ∣ r⟩ with r ∈ Φ(F) is Demushkin exactly when X is nonempty and the degree-one form of ⟦r⟧ is nondegenerate.

References

• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §3, Proposition 3.
• J.-P. Serre, Galois Cohomology, Chapter I, §4.5.
• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., Chapter III, §9.

theorem TauCeti.freeProP.cupFp_eq_zero_iff_degreeOneForm_eq_zero {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} [Finite X] {r : freeProP p X} {G : Type v} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hr : r ∈ proPFrattini p (freeProP p X)) (e : presentedProP p X {r} ≃ₜ* G) (a b : ↑(cohomFp p G 1).toModuleCat) : ((cupFp p G) a) b = 0 ↔ ((degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)) (((↑e).comp (ContinuousMonoidHom.quotientMk (Subgroup.normalClosure {r}).topologicalClosure)).continuousZModDualMap ((cohomFpLinearEquivContinuousZModDual p G) a))) (((↑e).comp (ContinuousMonoidHom.quotientMk (Subgroup.normalClosure {r}).topologicalClosure)).continuousZModDualMap ((cohomFpLinearEquivContinuousZModDual p G) b)) = 0

The cup form of a one-relator pro-p group is the degree-one form of its relator. Let F be the free pro-p group on a finite type X, let r ∈ Φ(F), and let G ≅ ⟨X ∣ r⟩. The cup product a ⌣ b of two classes of H¹(G, 𝔽_p) vanishes exactly when the degree-one form of the class of r in gr_1(F) vanishes at the characters of F obtained from a and b by composing with F → G.

theorem TauCeti.freeProP.isAlt_degreeOneForm_iff_forall_cupFp_self_eq_zero {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} [Finite X] {r : freeProP p X} {G : Type v} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hr : r ∈ proPFrattini p (freeProP p X)) (e : presentedProP p X {r} ≃ₜ* G) : (degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)).IsAlt ↔ ∀ (a : ↑(cohomFp p G 1).toModuleCat), ((cupFp p G) a) a = 0

The cup form is alternating exactly when the degree-one form of the relator is. For G ≅ ⟨X ∣ r⟩ with r ∈ Φ(F), the degree-one form of the class of r in gr_1(F) is alternating exactly when every cup square a ⌣ a on H¹(G, 𝔽_p) vanishes.

theorem TauCeti.IsDemushkin.nondegenerate_degreeOneForm {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} [Finite X] {r : freeProP p X} {G : Type v} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hr : r ∈ proPFrattini p (freeProP p X)) (e : presentedProP p X {r} ≃ₜ* G) (hG : IsDemushkin p G) : (freeProP.degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)).Nondegenerate

The relator of a Demushkin group has nondegenerate degree-one form (Labute, Proposition 3). Let G ≅ ⟨X ∣ r⟩ with r ∈ Φ(F) be a Demushkin group. Then the degree-one form of the class of r in gr_1(F) is nondegenerate: it is the cup form of G up to sign, and the cup form of a Demushkin group is nondegenerate.

theorem TauCeti.isDemushkin_of_nondegenerate_degreeOneForm {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} [Finite X] {r : freeProP p X} {G : Type v} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hr : r ∈ proPFrattini p (freeProP p X)) (e : presentedProP p X {r} ≃ₜ* G) [Nonempty X] (hnd : (freeProP.degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)).Nondegenerate) : IsDemushkin p G

Labute's criterion, sufficiency. Let F be the free pro-p group on a nonempty finite type X, let r ∈ Φ(F), and suppose the degree-one form of the class of r in gr_1(F) is nondegenerate. Then G ≅ ⟨X ∣ r⟩ is a Demushkin group: it is pro-p and topologically finitely generated, the relator functional of r is an isomorphism H²(G, 𝔽_p) ≅ 𝔽_p, and the cup form of G is the degree-one form of ⟦r⟧ up to sign, hence nondegenerate.

theorem TauCeti.isDemushkin_iff_nondegenerate_degreeOneForm {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} [Finite X] {r : freeProP p X} {G : Type v} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hr : r ∈ proPFrattini p (freeProP p X)) (e : presentedProP p X {r} ≃ₜ* G) : IsDemushkin p G ↔ Nonempty X ∧ (freeProP.degreeOneForm (gradedMk p (freeProP p X) 1 ⟨r, ⋯⟩)).Nondegenerate

Labute's criterion. Let F be the free pro-p group on a finite type X, let r ∈ Φ(F), and let G ≅ ⟨X ∣ r⟩. Then G is a Demushkin group exactly when X is nonempty and the degree-one form of the class of r in gr_1(F) is nondegenerate.