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What if you define a topology like the Whitney C^infty topology, but instead of the usual topology on the jet bundles, every fiber above dimension 1 has the indiscrete topology. Then this defines a courser topology than the usual one. To me it looks like it still gives a homotopy equivalence from embeddings to the frame bundle (since that part of the topology is preserved).
@AchimKrause what happens if you attach a 3-cell to $\Omega S^2 \wedge S^2$ by $(h, 2)$ where "h" is the adjoint of the hopf map. This is an unstable analog of what you suggest. Does it split in this case?
@WillSawin If by product you mean tensor product, then the resulting window element will still square to zero. If you meant direct product/sum, then you can do that but it is not so interesting since the algebra is decomposable.
