On the Continuous Cohomology of a semi-direct product Lie group

Naoya Suzuki

Published 2018-04-05, updated 2018-07-18Version 2

Let $G$ be a Lie group and $H$ be a subgroup of it. We can construct a bisimplicial manifold $NG(*) \rtimes NH(*)$ and the de Rham complex $\Omega^*(NG(*) \rtimes NH(*))$ on it. This complex is a triple complex and the cohomology of its total complex is isomorphic to $H^*(B(G \rtimes H))$. In this paper, we show that the total complex of the double complex $\Omega^q(NG(*) \rtimes NH(*))$ is isomorphic to the continuous cohomology $H_c^*(G \rtimes H;S^q{\mathcal G} \times S^q{\mathcal H})$ for any fixed $q$.