Are there theorems about mathematical objects that are later shown to be nonexistent?

Question

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.

Answer 1

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.