I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be established. Here is an example of a theorem that as far as I am aware fits into this category:
Theorem (Sullivan). A topological manifold of dimension $n\neq 4$ admits a unique Lipschitz structure.
The original paper of Sullivan [S] is a brief 13 pages note published in a conference proceeding. It contains a plethora of ideas, but hardly any proofs. Nevertheless the paper is a widely recognized as a source for the proof of the above result.
I am not the only one who struggled with this paper. Tukia and Vaisala [TV] wrote a 40 pages long paper whose aim was to understand some of the arguments on Sullivan. However they wrote in the introduction:
Since the presentation in [S] is very sketchy, a large part of this article is devoted to a fairly detailed exposition of Sullivan's theory. We take on faith the most difficult part, namely the existence of Sullivan groups.
I am reluctant to accept faith as a mathematical argument and so I am reluctant to accept that the paper [S] contains the proof of the theorem, so unless there is a detailed proof somewhere else, I think the above result is an example of a theorem that has not been proved yet. If I am wrong, please provide correct references.
Question. What are the other examples of such results?
[S] Sullivan, D.: Hyperbolic geometry and homeomorphisms. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 543–555, Academic Press, New York-London, 1979.
[TV] Tukia, P.; Väisälä, J.: Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982).