I am interested in learning if there have ever been such cases before. As a guide, I heard somewhere about a doctorate dissertation that allegedly proves some theorems about a kind of four-dimensional manifolds, only to be shown later that the four-dimensional manifolds satisfying those preconditions do not exist. I could not recall if it is just mere hearsay, and would appreciate if anyone knows of any similar stories.

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    $\begingroup$ Several stories of that flavor can be found here. $\endgroup$ – angryavian May 12 '17 at 2:44
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    $\begingroup$ There are several theorems about the field with one element. Unfortunately, it was known not to exist before the theorems were proven, so it probably doesn't quite qualify. On the plus side, linear combinations of vectors over this field are quite easy to deal with (see what I did there?). $\endgroup$ – David Wheeler May 12 '17 at 2:55
  • $\begingroup$ @JonasMeyer Yes, I meant hearsay. Sorry for the typo. Thank you. $\endgroup$ – discretizer May 12 '17 at 3:42
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    $\begingroup$ There is nothing controversial about the field with one element. The idea that it is an actual field is what is confused and uninformed, much as quantum groups are not groups and many other instances of the same phenomenon. $\endgroup$ – Mariano Suárez-Álvarez May 12 '17 at 5:09
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    $\begingroup$ You make it sound like a big waste of time, but the way you prove that something doesn't exist is by proving stuff about the nonexistent thing until you get a contradiction. I believe quite a bit was known about projective planes of order $10$ before they were shown not to exist. People have proved various things about odd perfect numbers, but it won't be a big letdown when they are finally proved not to exist. $\endgroup$ – bof May 12 '17 at 6:22

I'm not sure this falls under the category of "similar stories," but I think it's interesting and relevant:

Your question seems to focus on the negative aspect - however, sometimes results about nonexistent objects are really useful.

For a very good example of this, look at the proof of Fermat's last theorem. Ultimately, this was proved by passing through a seemingly unrelated area - namely, by proving the Taniyama-Shimura conjecture, a statement about polynomials of the form $y^2=x^3+ax+b$ (or rather, the curves they describe). Roughly speaking, Taniyama-Shimura states that all elliptic curves have a certain nice property called "modularity" - more precisely, every elliptic curve is "equivalent" in a precise sense to one which comes from a kind of function called a modular form in a specific way. (For those familiar with fields, insert the words "over $\mathbb{Q}$" everywhere in the previous couple sentences.)

To an outsider like me, elliptic curves have no obvious connections with Fermat, and modular forms have even fewer - so why was Taniyama-Shimura important here? The answer is the set of Frey curves, elliptic curves gotten from the hypothetical counterexamples to Fermat. Frey conjectured, and Ribet (following Serre) proved, that the Frey curves were nonmodular (if they existed) - by proving Taniyama-Shimura, Wiles showed that the Frey curves could not exist and hence proved Fermat's last theorem. Had it not been for Ribet's result about these nonexistent objects, this wouldn't have worked!

Note: Ribet didn't just prove a fact about nonexistent objects, he proved a very powerful theorem about elliptic curves in general. I'm focusing on one particular consequence of that theorem, which was a nontrivial subtheorem about a nonexistent object.

For another example, but so far above my pay grade that I can't begin to give a good explanation, there was serious work done on a particular object called the "Smith-Toda complex," a large subclass of which turned out to not be that complex after all. (OK fine, for $p>5$.) Interestingly for the context of this question, because of my aforementioned lack of knowledge I don't actually know whether this was ultimately a positive result a la Ribet or a negative result killing off a class of desired structures - tentatively towards the latter, I do believe that the existence of Smith-Toda complexes would have had a positive role in unstable homotopy theory. I happened to overhear about this while walking through the math department a while ago, and this question jogged my memory.

An interesting inverse to the question asked is the work of Jack Silver in set theory. My understanding is that much (all?) of Silver's work on the combinatorics of measurable cardinals was with the goal of proving their nonexistence; at present, among set theorists measurables are widely believed to be consistent, partly due to the insights provided by Silver's work.

EDIT: A much better example of this is early work on hyperbolic geometry, see Eric Wofsey's comment below.

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  • $\begingroup$ To provide a bit more context on Smith-Toda complexes: a Smith-Toda complex $V(k)$ depends on two parameters, a natural number $k$ and a prime $p$. They exist only when $k$ is small relative to $p$. The paper you linked shows that for $k=(p+1)/2$, $V(k)$ does not exist (it states this only for $p>5$, but that is because the cases $p\leq 5$ were already known previously). Smith-Toda complexes are "desired" in that their existence roughly means that homotopy theory is more "algebraic", and thus easier to understand. $\endgroup$ – Eric Wofsey May 12 '17 at 5:49
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    $\begingroup$ Another famous "inverse" example similar to Silver's work on measurable cardinals is the early work on hyperbolic geometry, which was done by people trying to prove it was inconsistent! $\endgroup$ – Eric Wofsey May 12 '17 at 5:53
  • $\begingroup$ @EricWofsey Thanks for the context! I've edited to clarify that it's only a certain class of ST-complexes which were shown not to exist. (My understanding from a cursory google is that they were first shown to not be ring spectra, and then shown to not exist at all; is this accurate?) Re: hyperbolic geometry, of course that's a much better example, don't know why I didn't think of it; I've edited it in. $\endgroup$ – Noah Schweber May 12 '17 at 6:08

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