Cohomology of Local Cochains

Details Cohomology of Local Cochains Cohomology of Local Cochains

2011-10-12

arxiv.org/abs/1110.2661v2

Mathematics

Algebraic Topology

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

Scientific paper

We prove that for generalised partitions of unity ${\phi_i \mid i \in I}$ and coverings $\mathfrak{U}:={\phi_i^{-1} (R \setminus {0}) \mid i \in I}$ 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 $G$.

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