Cohomology of Local Cochains

Abstract

We prove that for generalised partitions of unity ${\varphi }_i\mid i\in I$ and coverings $\mathfrak{U}:={\varphi }_i^{-1}(R\setminus 0)\mid i\in I$ of a topological space $X$ the cohomology of abstract $\mathfrak{U}$-local cochains coincides with the cohomology of continuous $\mathfrak{U}$-local cochains, provided the coefficients are loop contractible. Furthermore we show that for each locally contractible group $G$ and loop contractible coefficient group $V$ the complex of germs of continuous functions on left-invariant diagonal neighbourhoods computes the Alexander-Spanier and singular cohomology; similar results are obtained for $k$-groups and for germs of smooth functions on Lie groups $G$.

38 pages, LaTeX, elucidated argument for Corollaries 2.8 and 4.5, added Corollary 4.6 and examples in section 4, results unchanged

Cited by in corpus (1)

• Topological Group Cohomology with Loop Contractible Coefficients