2022/02/19 by Lukas Miaskiwskyi, Miaskiwskyi, Lukas
Mathematics · Medicine · #57R32 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2202.09666
openalex publication_date 2022/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.