2023/08/16 by Constantin-Nicolae Beli, Beli, Constantin-Nicolae
Mathematics · #20J06 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2308.08368
openalex publication_date 2023/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a group G and a G-module M, we denote by (C(G,M),d) the corresponding cochain complex obtained from the standard resolution. An element of the cohomology H(G,M) will be written as the class [a] of some cocycle a∈ C(G,M). The first result involves the triviality of the action of G on H(G,M), i.e. s[a]=[a] ∀ [a]∈ Hn(G,M), s∈ G. Adapted to cochains, we prove that sa-a=(hsd+dhs)(a) ∀ a∈ Cn(G,M), for some explicit map hs:C(G,M)→ C(G,M)[-1]. The second result regards the commutativity of the cup product, i.e. [a]∪ [b]=(-1)pqt_*([b]∪ [a]) ∀ [a]∈ Hp(G,N), [b]∈ Hq(G,M). (Here t:N⊗ M→ M⊗ N is the natural bijection.) Adapted to cochains, we prove that (-1)pqt_*(b∪ a)-a∪ b=(hd+dh)(a⊗ b) ∀ a∈ Cp(G,M), b∈ Cq(G,N), for some explicit map h:C(G,M)⊗ C(G,N)→ C(G,M⊗ N)[-1].