Abstract: The plan is to make some useful homotopical computations in spaces such as $$\mathrm{TOP}(n)/\mathrm{O}(n)$$, which is important in smoothing theory, or the closely related structure space of the n-sphere, $$G(n+1)/\mathrm{TOP}(S^n)$$. The method is to ask how (or whether) topological automorphisms or homotopy automorphisms of a sphere act on configurations of very low cardinality in the sphere. It is an old-fashioned investigation, but there are some connections to functor calculus in different shades.