A note on relative Gelfand–Fuks cohomology of spheres

Abstract We study the Gelfand–Fuks cohomology of smooth vector fields on relative to following a method of Haefliger that uses tools from rational homotopy theory. In particular, we show that injects into the relative Gelfand–Fuks cohomology which corrects a claim by Haefliger. Moreover, for the relative Gelfand–Fuks cohomology agrees with the smooth cohomology of and we provide a computation in low degrees.