A note on relative Gelfand-Fuks cohomology of spheres

2025/03/10 by Prigge, Nils
#57R32 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.07322

Abstract

We study the Gelfand-Fuks cohomology of smooth vector fields on S^d relative to SO(d+1) following a method by Haefliger that uses tools from rational homotopy theory. In particular, we show that H^*(BSO(4);ℝ) injects into the relative Gelfand-Fuks cohomology which corrects a claim by Haefliger. Moreover, for S³ the relative Gelfand-Fuks cohomology agrees with the smooth cohomology of Diff⁺(S³) and we provide a computation in low degrees.

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