Continuous Hu cohomology
Let \(X\) be a topological space and \(\Omega_X\) the set of all open coverings of \(X\). Let \(G\) be a topological abelian group. The group of all continuous functions \(\varphi: X^{q+1}\rightarrow G\) is denoted by \(\Phi^q(X,G)\) and the subgroup of \(\Phi^q(X,G)\) consisting of all locally zero functions defined in terms of open coverings of \(X\) is denoted by \(\Phi^q_0(X,G)\). Here a locally zero functions means a continuous map \(\varphi: X^{q+1}\rightarrow G\) such that there is an open cover \(\alpha\) with \(\varphi(\alpha^{q+1})=g_0\), the identity of \(G\) and \(\alpha^{q+1}=\bigcup {U_{\alpha}}^{q+1}\), \(U_{\alpha}\in \alpha\). The quotient \(\Phi^q(X,G)\mid \Phi^q_0(X,G)\) is denoted by \(\overline{\Phi}^q(X,G)\). Defining boundary homomorphisms \(\delta : \Phi^{q-1}(X,G)\rightarrow \Phi^q(X,G)\) via the projection maps, the author obtains a cochain complex \(\Phi^\star(X,G)\) and then obtains a quotient cochain complex \(\overline{\Phi}^\star(X,G)\) by the reduced boundary maps \(\delta: \Phi_0^{q-1}(X,G)\rightarrow \Phi_0^q(X,G)\). The cohomology of the cochain complex \(\Phi^\star(X,G)\) is denoted by \(H^\star(X,G)\) and called \textit{continuous Hu cohomology} of \(X\). A function \(\varphi: X^q\rightarrow G\) is called \textit{partially continuous} if there exists \(\alpha\in \Omega_X\) such that the restriction \(\varphi\mid \alpha^q\) is a continuous map. Alexander-Spainer cohomology was given in [\textit{L. Mdzinarishvili}, ``Partially continuous Alexander-Spanier cohomology theory'', Preprint-Reihe, Topologie und Nichtkommutative Geometrie, Mathematisches Institut, Universität Heidelberg, Heft No. 130 (1996)] in terms of these partially continuous maps. In this paper these two cohomologies are compared and some connections between them determined.
scientific article; zbMATH DE number 6053763