A Spectral Sequence Connecting Continuous With Locally Continuous Group Cohomology

We present a spectral sequence connecting the continuous and ’locally continuous’ group cohomologies for topological groups. As an application it is shown that for contractible topological groups these cohomology concepts coincide. Similar results for k-groups and smooth cochains on Lie groups are also obtained.

💡 Research Summary

The paper investigates two cohomology theories attached to a topological group (G) with a continuous (G)-module (V): the classical continuous group cohomology (H_c^(G;V)), built from cochains that are globally continuous, and a “locally continuous” cohomology (H_{cg}^(G;V)), where cochains are required to be continuous only on some neighbourhood of the diagonal in (G^{n+1}). The latter is sometimes called locally continuous group cohomology because it relaxes the continuity condition, allowing more cochains and often yielding different cohomology groups. A well‑known example is the compact Hausdorff group (G=\mathbb{R}/\mathbb{Z}) with integer coefficients, where the two theories give distinct results.

The central goal is to relate these two theories via a spectral sequence. The author works in the general setting of a transformation group ((G,X)) together with a (G)-invariant open covering (\mathcal U) of the space (X). For each covering one defines a bi‑graded abelian group \