A note on relative Gelfand-Fuks cohomology of spheres
Nils Prigge
TL;DR
The paper studies the relative Gelfand–Fuks cohomology of smooth vector fields on spheres relative to $SO(d+1)$ using Haefliger’s rational-homotopy framework. It constructs explicit relative Sullivan models for the relevant Borel fibrations and section spaces, correcting Haefliger’s kernel description in the even-$d$ case and proving a refined injection for $d=3$ involving a new class $\bar{z}_1$. For $d=3$ it provides a concrete low-degree computation of the smooth cohomology of $\mathrm{Diff}_+(\mathbb{S}^3)$ via a Van-Est isomorphism, including an explicit relative model and differentials up to degree $12$. The results connect the algebraic structure of $H^*(\mathcal{L}_{\mathbb{S}^d};SO(d+1))$ to the cohomology of diffeomorphism groups, aligning with Nariman’s and Pri24I’s findings on Euler and Pontryagin data and offering explicit computational tools for future work.
Abstract
We study the Gelfand-Fuks cohomology of smooth vector fields on $S^d$ relative to $\mathrm{SO}(d+1)$ following a method by Haefliger that uses tools from rational homotopy theory. In particular, we show that $H^*(\mathrm{BSO}(4);\mathbb{R})$ injects into the relative Gelfand-Fuks cohomology which corrects a claim by Haefliger. Moreover, for $S^3$ the relative Gelfand-Fuks cohomology agrees with the smooth cohomology of $\text{Diff}^+(S^3)$ and we provide a computation in low degrees.