The plan is to make some useful homotopical computations in spaces such as  TOP(n)/O(n), which is important in smoothing theory, or the closely related structure space of the n-sphere, G(n+1)/TOP(n-sphere). 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.

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