English
For x in cocycles₂(A), H2π A x = 0 if and only if x is a coboundary.
Русский
Для x в cocycles₂(A) выполняется, что H2π A x = 0 тогда и только тогда, когда x — кособдуманный ( coboundary).
LaTeX
$$$H2\pi_A(x) = 0 \Leftrightarrow x \in coboundaries_2(A)$$$
Lean4
theorem H2π_eq_zero_iff (x : cocycles₂ A) : H2π A x = 0 ↔ ⇑x ∈ coboundaries₂ A :=
by
have h :=
leftHomologyπ_naturality'_assoc (isoShortComplexH2 A).inv (shortComplexH2 A).moduleCatLeftHomologyData
(leftHomologyData _) ((inhomogeneousCochains A).sc 2).leftHomologyIso.hom
simp only [H2π, isoCocycles₂, π, HomologicalComplex.homologyπ, homologyπ, cyclesMapIso'_inv, leftHomologyπ, ← h,
← leftHomologyMapIso'_inv, ModuleCat.hom_comp, LinearMap.coe_comp, Function.comp_apply,
map_eq_zero_iff _ ((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
simp [LinearMap.range_codRestrict, coboundaries₂, shortComplexH2, cocycles₂]
