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Seiberg-Witten Theory, thread 1/2: 4-manifolds and "why Seiberg-Witten Theory?" After this I will do a thread going a little more into the technical details, because I think too often the analysis is glossed over and it& #39;s actually really cool! 1/n
This thread won& #39;t be too technical, but I& #39;m going to in general be assuming familiarity with connections on principal bundles and the basic ideas of algebraic topology. This will be more important in the next thread. 2/n
The story starts with the question of understanding smooth manifolds. Perhaps two motivating questions are: do all manifolds have a smooth structure? and do manifolds have a unique smooth structure? For 2-manifolds, the answer to both is yes and has been known forever. 3/n
With a little more sophisticated machinery one can prove the same is true for 3-mfds. The first counterexample came in 1960 when Kervaire gave an example of a 10-mfd with no smooth structure. In 1962 Smale proved the overpowered h-cobordism theorem for dimensions \geq 5 4/n
From which we can deduce that in dim \geq 5 there are manifolds with multiple smooth structures, but at least for compact manifolds there are always finitely many smooth structures. We can even count them sometimes, such as for spheres! And R^n has a unique smooth structure. 5/n
The proof of the h-cobordism theorem fails in 4 dimensions (not enough room for the "Whitney trick" to resolve critical points in a Morse function along your h-cobordism). So for a long time, the question lingered: is 4 dimensions more like 3 or 5+ dimensions? 6/n
The h-cobordism theorem gives straightforward conditions to prove that a certain type of cobordism yields a diffeomorphism. Without it, 4-mfd theorists were forced to work with very little. In particular, there were no invariants that contained smooth information 7/n
since the standard algebraic-topology methods don& #39;t see smooth structures. We had some minor results, such as Rokhlin& #39;s theorem which restricts the algebraic topology of certain 4-mfds, but no real tools. 8/n
Before I start telling you about the wild year of 1982, I want to talk a little bit about 4 mfds. The cup/wedge product combined with evaluation on the fundamental class gives a bilinear form on H^2(M, Z) (or H_2 by Poincare duality). This is the intersection form Q. 9/n
Q tells you a lot about your mfd, especially if it& #39;s simply connected. For example, for closed simply connected Q determines the Stiefel-Whitney, Euler, and Pontryagin classes, and Whitehead& #39;s theorem says two mfds with the same intersection form are homotopy equivalent. 10/n
Okay so Q is very very important. In 1982, two bombshells dropped. First was a paper by Michael Freedman which, among other advances, proved the h-cobordism theorem in the continuous category. 11/n
He showed that for any Q, there are either 1 or 2 topological manifolds with intersection form Q, thereby classifying compact simply connected topological 4-manifolds up to homeomorphism. 12/n
So we& #39;re looking pretty similar to 2 and 3 dimensions: each homotopy class has at most two homeomorphism classes. That& #39;s not too bad. 13/n
Second was a paper by Donaldson, using Yang-Mills gauge theory arising from physics to define new invariants which were sensitive to smooth structure. He also proved that for a smooth manifold with Q being definite, then Q must be diagonalizable. 14/n
Bam! By both results, we have a whole suite of 4-mfds that admit no smooth structure. There& #39;s question 1 for ya: it& #39;s now looking like 4 dimensions is gonna be pretty similar to 5+ dimensions. 15/n
Moreover, these new invariants quickly allowed for examples of manifolds with multiple smooth structures, resolving question 2. Great! Problem solved! Well...not quite. 16/n
Because then Cliff Taubes used Donaldson Theory to show that R^4 has uncountably many smooth structures. 17/n
In the next decade, old conjectures were resolved, and our understanding greatly improved. It became pretty clear that 4d is wild. But we were far from anything resembling a good understanding. One problem: Donaldson Theory was HARD. 18/n
The rough schema of Donaldson theory is as follows: write some equations, and look at the "moduli space" of all solutions, modulo some symmetries. This will form a smooth mfd. Topological invariants of this mfd will be smooth invariants of your base mfd. 19/n
Said equations, the Yang-Mills equations, are highly nonlinear, with nonabelian "gauge group," and a noncompact moduli space that one has to compactify before one can say anything. Computing the invariants was at best a pain and at worst hopeless. 20/n
Then came a paper by Seiberg and Witten in 1994, writing down a new set of equations for various physics-y purposes. And a whole new revolution happened. 21/n
The idea of Seiberg-Witten theory is the same as that of Donaldson theory, but with a few changes: the equations have milder nonlinearity, an abelian gauge group, and a compact moduli space. This makes the invariants much more computable and the analysis much friendlier. 22/n
More conjectures, such as the Thom conjecture, were quickly resolved, and it seemed (and was "proved" by the physicists) that Seiberg-Witten theory contains all the information of Donaldson theory. 23/n
In the 27 years since, Seiberg-Witten theory has continued to yield new advances and become more refined. However, there is still so much we don& #39;t know. 24/n
A snapshot: we have procured many examples of manifolds with infinitely many smooth structures. We have yet to find an example of a manifold with a unique, or even finitely many smooth structures. And the 4-sphere? We know absolutely nothing about the 4-sphere. 25/n
Stay tuned next time for a deeper dive into how the Seiberg-Witten invariants are actually constructed! 26/n, n = 26.
