A local-to-global analysis of Gelfand-Fuks cohomology

TL;DR

This paper introduces a new local-to-global method for analyzing Gelfand-Fuks cohomology using generalized covers, providing a unified framework that can extend to other geometric cohomology theories.

Contribution

It presents a novel proof technique for the Gelfand-Fuks spectral sequence and offers a comprehensive exposition on continuous Chevalley-Eilenberg cohomology of vector fields.

Findings

New local-to-global proof technique for Gelfand-Fuks cohomology

Unified approach to comparing sheaf-like data over manifold powers

Modernized exposition on cohomology of formal and Euclidean vector fields

Abstract

We present a novel proof technique to construct the Gelfand-Fuks spectral sequence for diagonal Chevalley-Eilenberg cohomology of vector fields on a smooth manifold, performing a local-to-global analysis through a notion of generalized good covers from the theory of factorization algebras and cosheaves. This approach yields a unified way to deal with the problem of comparing "sheaf-like" data over different Cartesian powers of the manifold, and is easily generalized to the study of other cohomology theories associated to geometric objects. Independently, we lay out a detailed and easily accessible exposition on the continuous Chevalley-Eilenberg cohomology of formal vector fields and of vector fields on Euclidean space, modernizing and elaborating on well-established literature on the subject by Bott, Fuks, and Gelfand.