On Topological Structure of the First Non-abelian Cohomology of Topological Groups
H. Sahleh *
Department Of Mathematics, Faculty Of Mathematical Sciences, University of Guilan, P. O. Box 1914, Rasht, Iran.
H. E. Koshkoshi
Department Of Mathematics, Faculty Of Mathematical Sciences, University of Guilan, P. O. Box 1914, Rasht, Iran.
*Author to whom correspondence should be addressed.
Abstract
Let G, R, and A be topological groups. Suppose that G and R act continuously on A, and G acts continuously on R. In this paper, we dene a partially crossed topological G - R-bimodule (A, μ), where μ: A→ R is a continuous homomorphism. Let Derc(G, (A, μ)) be the set of all (α, r) such that α : G → A is a continuous crossed homomorphism and μα(g) = rgr-1. We introduce a topology on Derc(G, (A, μ)). We show that Derc(G, (A, μ)) is a topological group, wherever G and R are locally compact. We define the first cohomology, H1(G, (A, μ)), of G with coecients in (A, μ)) as a quotient space of Derc(G, (A, μ)). Also, we state conditions under which H1(G, (A, μ)) is a topological group. Finally, we show that under what conditions H1(G, (A, μ)) is one of the following: k-space, discrete, locally compact and compact.
Keywords: Non-abelian cohomology of topological groups, Partially crossed topological bimodule, Evaluation map, Compactly generated group