Continuous cohomology of a trivial group vanishes in positive degrees

Let G be a topological group with at most one element and X a topological representation of G, with no topological hypothesis on X. Then Hⁿ⁺¹(G, X) = continuousCohomology (n + 1) X vanishes for every n. Evaluation at the identity contracts the coinduced resolution X → C(G, X) → C(G, C(G, X)) → ⋯ (TopRep.d_sum_apply_add_sum_d_apply at the single point 1), and over a trivial group every element of the resolution is invariant, so the contraction descends to the homogeneous cochains and exhibits every cocycle of positive degree as a coboundary.

Main results

• TauCeti.ContinuousCohomology.subsingleton_continuousCohomology_succ_of_subsingleton: Hⁿ⁺¹(G, X) vanishes when G is a subsingleton.

instance TauCeti.ContinuousCohomology.subsingleton_continuousCohomology_succ_of_subsingleton {k : Type u_1} {G : Type u_2} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Subsingleton G] (X : TopRep k G) (n : ℕ) : Subsingleton ↑(continuousCohomology (n + 1) X).toModuleCat

A trivial group has no continuous cohomology in positive degrees. For a topological group G with at most one element and any topological representation X of G, Hⁿ⁺¹(G, X) vanishes: evaluation at the identity contracts the coinduced resolution, and every element is invariant.