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Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some details making it difficult to understand, see: Well known theorems that have not been proved.

On the other hand Donaldson and Sullivan [DS] proved that there are homeomorphic smooth $4$-dimensional manifolds that are not bi-Lipschitz homeomorphic, so the Lipschitz structure of a $4$-manifold, even if it exists, need not be unique.

Question. Is the Lipschitz structure on $\mathbb{S}^4$ unique?

[DS] S. K. Donaldson, D. P. Sullivan,, Quasiconformal 4-manifolds. Acta Math. 163 (1989), 181-252.

[S] D. Sullivan, Hyperbolic geometry and homeomorphisms. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 543–555, Academic Press, New York-London, 1979.

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  • $\begingroup$ Could you make it more explicit which results here are for closed manifolds and w hich are for open manifolds? $\endgroup$
    – Mikhail Katz
    Commented Jul 25, 2023 at 12:37
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    $\begingroup$ I am quite sure that this (uniqueness on $S^4$) is unknown. $\endgroup$
    – Moishe Kohan
    Commented Jul 25, 2023 at 13:40
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    $\begingroup$ I think this was listed as an open problem at a recent BIRS 4-manifolds meeting. My understanding is Kirby is revising his problem list. When the revised version comes out you should have your answer, but my recollection from the meeting is it remains an open problem. $\endgroup$
    – Ryan Budney
    Commented Jul 26, 2023 at 21:20

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