Transgression for a minimal free pro-p presentation

Let F = freeProP p X and let R be a closed normal subgroup contained in its pro-p Frattini subgroup. For finite abelian coefficients M of exponent dividing p with trivial action, the restriction H¹(F, M) → H¹(R, M) is zero, while H²(F, M) vanishes. The five-term sequence therefore makes the transgression H¹(R, M) ^ (F ⧸ R) → H²(F ⧸ R, M ^ R) bijective. For a finite generating type, R ≤ Φ(F) characterizes minimal presentations (TauCeti.presentedProP.subset_proPFrattini_iff_card_eq). This is the first step toward interpreting dim H²(G, 𝔽_p) as the number of relations of G.

Main result

• TauCeti.freeProP.transgression_bijective: transgression is bijective for a closed normal subgroup R ≤ Φ(F) and finite abelian coefficients of exponent dividing p with trivial action.

References

• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.9.5).
• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), §1.4.
• J.-P. Serre, Galois Cohomology, Chapter I, §4.3.

theorem TauCeti.freeProP.transgression_bijective {p : ℕ} [Fact (Nat.Prime p)] {X : Type u} {M : Type v} [CommGroup M] [TopologicalSpace M] [DiscreteTopology M] [Finite M] [MulDistribMulAction (freeProP p X) M] [ContinuousSMul (freeProP p X) M] (R : Subgroup (freeProP p X)) [R.Normal] (hRc : IsClosed ↑R) (hR : R ≤ proPFrattini p (freeProP p X)) (htriv : ∀ (g : freeProP p X) (m : M), g • m = m) (hexp : ∀ (m : M), m ^ p = 1) : Function.Bijective ⇑(ContCohomology.transgression (freeProP p X) (Additive M) R hRc)

The transgression of a minimal presentation is an isomorphism. Let F = freeProP p X and let R be a closed normal subgroup of F contained in its pro-p Frattini subgroup, as for the relation subgroup of a minimal presentation. For a finite abelian group M of exponent dividing p with trivial F-action, for instance 𝔽_p, the transgression H¹(R, M) ^ (F ⧸ R) → H²(F ⧸ R, M ^ R) is bijective.