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I remember coming across some article of Bill Thurston’s where he describes a 3-manifold (with boundary?) as being like an egg. In my recollection the interior of the egg, the shell, and even the shell after being cracked all had meaning. As you can imagine, Googling “like an egg three-manifold Thurston” is not very fruitful. If you happen to remember this article’s title, please point me in the right direction!

If I had to guess, I would imagine that this article is more likely to be a research announcement or notes of some kind rather than a paper proving a major result. I guess this because the paper was somewhat “talk-y”. I hope that this paper really exists out there and I haven’t just imagined it!

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I believe you're looking for page 211 of Hyperbolic Structures on 3-Manifolds I: Deformation of Acylindrical Manifolds (Annals of Math., 1986), which includes the following paragraph:

A complete description of the three spaces $AH(M)$, $GH(M)$, and $QH(M)$ is certainly not rigorously known, but here is a conjectural image, of which certain features can be rigorously proved. Let us stick to the case that $M$ is a compact, acylindrical manifold. Then $H(M)$ is a hard-boiled egg. The egg complete with shell is $AH(M)$; it appears to be homeomorphic to a closed unit ball. $GH(M)$ is obtained by thoroughly cracking the egg shell on a convenient hard surface. Apparently no material is physically separated from the egg, but many cracks are developed - cracks are dense in the boundary - and at the same time, the material of the egg just inside the shell is weakened, so that neighborhood systems of points on the boundary become thinner. Finally, $QH(M)$ has uncountably many components, which are obtained by peeling off the shell and scattering the pieces all over. Each component is homeomorphic to some Teichmuller space - it is parametrized by Euclidean space of some even dimension. "Most" of the components have dimension zero, for they describe groups whose limit set is all of $S_\infty^2$.

Here $H(M)$ is the set of complete hyperbolic manifolds $N$ with a homotopy equivalence $f:M\to N$, $AH(M)$ is $H(M)$ equipped with the 'algebraic topology', $GH(M)$ with the 'geometric topology', and $QH(M)$ with the 'quasi-isometric topology'.

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    $\begingroup$ Which is the first paper obtained by googling "William Thurston 3-manifold egg"... $\endgroup$
    – Chris Gerig
    Commented Nov 20, 2019 at 16:19
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    $\begingroup$ @ChrisGerig ahh, maybe somehow it was the William I was missing. $\endgroup$
    – Robbie Lyman
    Commented Nov 20, 2019 at 16:25
  • $\begingroup$ Thank you, that was very fast! $\endgroup$
    – Robbie Lyman
    Commented Nov 20, 2019 at 16:25
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    $\begingroup$ @ChrisGerig : Depends on what training Google's cookies in your browser have associated with you. I have a browser that does not know I study topology and its first hit is this MO question, followed by the MSRI page of Thurston's "Geometry and Topology of three-manifolds". There's a reasonable chance that various people's searches on this topic in the last 12 hours have also reweighted Google's results sent to my memoryless browser. $\endgroup$
    – Eric Towers
    Commented Nov 21, 2019 at 2:05
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    $\begingroup$ This website is amazing ^^ $\endgroup$
    – G. Fougeron
    Commented Nov 21, 2019 at 11:03

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