2018/04/03 by Kayvan Nejabati Zenouz, Zenouz, Kayvan Nejabati
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1804.01029
openalex publication_date 2018/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a topological group G and a Hausdorff topological group A on which G acts continuously and compatibly with the group operation of A , we study the set of continuous cocycles of G with value in A . This set is a function space and can be endowed with several topologies. By imposing a suitable function space topology on the set of cocycles of G with value in A , we propose a topological study of this set, and we prove, as our first main result, that if A is a compact group having a presentation as an inverse limit of compact and Hausdorff topological groups Ar , for r in a directed poset R , on which G acts continuously and compatibly with the group operation of Ar and equivariantly with respect to the transition maps, then one has a natural identification between the first nonabelian cohomology set of G with coefficients in the inverse limit A and the inverse limit of first nonabelian cohomology sets of G with coefficients in Ar . Furthermore, we prove, as our second main result, that if G is compact and Hausdorff, and A is abelian - therefore one can define cohomology groups for all n≥1 - then under a certain condition on Ar one has a natural identification between cohomology group with coefficients in the inverse limit A and the inverse limit of cohomology groups with coefficients in Ar , for all n≥1 .