Topological group cohomology with loop contractible coefficients (Q428788)

A topological group \(A\) is called loop contractible if there exists a contraction homotopy map \(H: A\times [0,1] \rightarrow A\) such that for each \(t\in [0,1]\) the map \(H(-,t) : A \rightarrow A\) is a group homomorphism. In this work the authors prove that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and locally continuous group cochains are isomorphic. They obtain similar results for compactly generated groups and finite dimensional Lie groups. They prove:NEWLINENEWLINEIf \(G\) is a Lie group whose finite products are smoothly paracompact and \(A\) is a smoothly loop contractible smooth \(G\)-module then the cohomology groups of smooth group cochains and locally smooth group chains are isomorphic.