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Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually interesting. To prove a property $P$ (largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc.) for a class of objects $X$ (e.g. fundamental groups of finite volume hyperbolic $3$-manifolds), embed each object of $X$ nicely in an object of a `universal receiver' class of objects $R$ (Right-Angled Artin Groups, or RAAGs), each of which are simple and has good properties. The existence of such an embedding in itself implies property $P$ which you are interested in, maybe with some additional effort (Agol's fibering theorem, tameness, etc.).

Proving a mathematical statement in this way makes a lot of sense, but I don't recall having seen this proof pattern before anywhere else in mathematics. Well, that's not entirely true- Cayley's Theorem that a group embeds in a permutation group has some corollaries (e.g. Given a group $G$ and subgroup $H$ with $[G:H]=n$, there exists a exists a normal subgroup $N$ of $G$, with $N\subseteq H$ such that $[G:N]|n!$).

Question: Which other conjectures, that objects in a class $X$ have a property $P$, have been proven by embedding objects in $X$ nicely as subobjects of objects in a universal receiver $R$ whose good properties imply $P$ for objects in $X$?

For a compelling example, it would have to be difficult to prove $P$ for objects in $X$ in any other way. For an even more compelling example, the universal receiver $R$ would be surprising (RAAGs are a surprising universal receiver, I think).

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Are you asking for conjectures or proof techniques? The answers so far refer to the latter. –  Martin Brandenburg Jul 1 '12 at 14:19
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6 Answers 6

Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that every finite-dimensional Lie algebra embeds into the Lie algebra of a general linear group. But once Ado's theorem is in place, the claim is quite easy: the Lie group is the space of curves in the general linear group that are tangent to the embedded linear algebra, up to smooth deformation.

(Without Ado's theorem, it is relatively straightforward to make the Lie algebra the tangent space of a local Lie group via the Baker-Campbell-Hausdorff formula, but establishing the global associative law necessary to extend this local Lie group to a global one is highly non-trivial. The main contribution of Ado's theorem, then, is allowing one to exploit the global associativity of the general linear group, which is a triviality in this concrete setting, but not in the setting of the abstract local group given by BCH.)

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This is exactly the kind of proof I was looking for- thanks! –  Daniel Moskovich Jul 2 '12 at 16:21
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Kodaira embedding theorem for complex manifolds. In this case the complex projective space (of some dimension) is the receiver. Once you have an embedding, you can use the entire machinery of algebraic geometry to study your complex manifold.

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This is not quite the same but it reminds me of Mitchell's embedding theorem.

This says that any small Abelian category is equivalent to a full subcategory of a category of $R$-modules. In particular, many facts about Abelian categories are can be reduced to questions of categories of $R$-modules. In particular, you can use diagram chases.

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So the OP mentioned corollaries of the Cayley Embedding Theorem, and here we have corollaries of the analogous Mitchell Embedding Theorem. –  Toby Bartels Jul 2 '12 at 8:10
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It is very common in set theory to prove that a particular model or structure is well-founded by mapping it into a fixed well-founded structure. The point is that if $j:\langle M,{\in^M}\rangle\to \langle N,{\in}\rangle$ is $\in$-preserving, then any instance of ill-foundedness in $M$ would carrry over to $N$, which has none; and so $M$ is well-founded. This method is often applied in the context of iterated ultrapower constructions.

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Part of the proof of the Green-Tao theorem involves embedding the primes into the "semiprimes", which are sufficiently pseudorandom for results like Szemerédi's theorem to apply. In fact, a lot of recent progress has been made with regards to the relative Szemerédi theorem, and maybe there is some hope that Erdős' conjecture on arithmetic progressions could be proven by embedding any sufficiently dense set of numbers into some pseudorandom set.

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This is perhaps an inferior example to those already mentioned, but my understanding of the so-called Grothendieck double-limit criterion (used to test if a map from a Banach space $X$ to a dual Banach space $Y^*$ is weakly compact) is that the proof proceeds along the following lines:

  • if $X$ and $Y$ are Banach spaces we view them as closed subspaces of $C(\Omega_X)$ and $C(\Omega_Y)$ where $\Omega_X$ is the closed unit ball of $X^*$ equipped with the weak-star topology, and likewise for $\Omega_Y$;

  • taking a bilinear map $X\times Y \to {\bf C}$, we appeal to Hahn-Banach to lift/extend it to a bilinear map $C(\Omega_X)\times C(\Omega_Y)\to {\bf C}$

  • now we are in a position to exploit things like Riesz representation, positivity, and ideas/tools from measure theory.

As will be seen from the sketchiness of this account, I may have misunderstood or misremembered what is going on. But the gist of my claim is that a non-trivial theorem about Banach spaces is proved by embedding a given Banach space into one which seems nicer, namely $C(\Omega)$.

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