[Paper Review] Cohomology of Local Cochains

Martin Fuchssteiner|arXiv (Cornell University)|Oct 12, 2011

Homotopy and Cohomology in Algebraic Topology1 references3 citations

TL;DR

This paper establishes that for loop contractible coefficient groups and coverings by cozero sets of generalized partitions of unity, the cohomology of abstract local cochains coincides with that of continuous local cochains. It further proves that for locally contractible topological groups and Lie groups, left-invariant diagonal neighbourhoods suffice to compute Alexander-Spanier and singular cohomology, extending Van Est's result to continuous and smooth cochains via a novel homotopy-theoretic approach.

ABSTRACT

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

Motivation & Objective

• To establish conditions under which abstract local cohomology agrees with continuous local cohomology.
• To extend Van Est's result on Alexander-Spanier cohomology in locally contractible topological groups to continuous and smooth cochains.
• To show that left-invariant diagonal neighbourhoods in Lie groups compute singular and Alexander-Spanier cohomology when coefficients are loop contractible.
• To unify Čech, singular, and Alexander-Spanier cohomology theories in the context of locally contractible topological groups and Lie groups.

Proposed method

• Uses generalized partitions of unity to define coverings by cozero sets, forming the basis for local cochain complexes.
• Introduces the Čech-Alexander-Spanier double complex to relate abstract, continuous, and Čech cochains.
• Applies homotopy-theoretic techniques, particularly the concept of loop contractibility, to show cochain complex quasi-isomorphisms.
• Employs colimit constructions over numerable coverings to recover classical cohomology theories for paracompact spaces and smoothly paracompact manifolds.
• Analyzes cohomology on topological and Lie groups using left-invariant diagonal neighbourhoods.
• Relies on the k-space and kω-space framework to ensure topological compatibility in colimit constructions.

Experimental results

Research questions

• RQ1 Under what conditions does the cohomology of abstract local cochains coincide with that of continuous local cochains?
• RQ2 Can left-invariant diagonal neighbourhoods in locally contractible topological groups compute Alexander-Spanier cohomology?
• RQ3 How do smooth local cochains relate to singular and Alexander-Spanier cohomology on Lie groups with smooth partitions of unity?
• RQ4 To what extent do loop contractible coefficients ensure equivalence between different cohomology theories?
• RQ5 What role do generalized partitions of unity play in unifying Čech, singular, and Alexander-Spanier cohomology?

Key findings

• For any covering by cozero sets of a generalized partition of unity and loop contractible coefficients, the cohomology of abstract local cochains is isomorphic to that of continuous local cochains.
• In locally contractible topological groups, the cohomology computed via continuous local cochains on left-invariant diagonal neighbourhoods agrees with Alexander-Spanier and singular cohomology.
• For Lie groups with smooth partitions of unity and smoothly loop contractible coefficients, smooth local cochains compute the same cohomology as singular and Alexander-Spanier theories.
• The colimit over all numerable coverings recovers classical Alexander-Spanier and continuous cohomology for paracompact spaces.
• The Čech-Alexander-Spanier double complex provides a bridge between abstract, continuous, and Čech cohomologies under the stated conditions.
• The results generalize Van Est's theorem by extending it to continuous and smooth cochains on topological and Lie groups.