A note on relative Gelfand-Fuks cohomology of spheres

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.