Theorems that led to very successful research programs in Geometry and Topology [closed]

Question

In the recent times I have heard a lot about the following:

1. The Atiyah-Singer Index theorem
2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications.

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.

Edit: With my thin background in geometry and topology I cannot make the question more precise or add more examples but I can tell why I would like to know such programs. After I heard these two theorems and their applications I thought every mathematician should know a little about these results even if they don't try to completely understand its proof. For example, Hamilton's 1982 result mentioned by PVAL is something I want to read about. Thanks for many good answers. Again, I emphasize that I have a very thin background in these things.

Answer 1

In 1982, Richard Hamilton proved that a smooth closed 3-manifold admitting a metric with strictly positive Ricci curvature is a spherical space form (in particular such a manifold which is simply connected is $S^3$). This led to a (by all accounts still active) research program culminating in Grigori Perelman's proof of Thurston's geometrization conjecture in 2006 (and hence the Poincare conjecture).

Answer 2

Donaldson's theorem uses the theory of instantons from mathematical physics to impose algebraic constraints on the possible intersection forms that a smooth 4-manifold can have (roughly speaking, they have to be diagonalizable over the integers). But Freedman had shown that there are a great many topological 4-manifolds whose intersection forms do not satisfy Donaldson's constraints, and consequently there are a lot of topological 4-manifolds which do not admit smooth structures.

This led to vigorous ongoing activity in low dimensional topology. Seiberg-Witten theory simplified Donaldson's argument by replacing instantons with monopoles, and this theory gives most of the state-of-the art results in the area of exotic 4-manifolds. Floer used Donaldson's instantons to build a homology theory associated to a 3-manifold, and Donaldson noticed that a cobordism between 3-manifolds induces a map between their instanton homology groups, ultimately leading Atiyah and others to write down the axioms for what we now know as TQFT. Floer's idea was adapted to construct other homology theories for 3-manifolds, contact manifolds, symplectic manifolds, etc. and these tools are still proving state-of-the-art theorems in their areas.

Answer 3

Let $G_1 \subset G_2 \subset \dots$ be a sequence of groups and inclusion homomorphisms. It satisfies homological stability if for every $r \in \mathbb{N}_0$ there is an $n(r)$, such that for all $n > n(r)$ the homomorphism $$ H_r(G_n) \to H_r(G_{n+1}) $$ induced by the inclusion is an isomorphism. Examples of groups that have homological stability are the symmetric groups $S_n$, the braid groups $B_n$ and the automorphism group of the free groups $Aut(F_n)$.

Harer proved that the mapping class groups $\Gamma_{g,b}$ of surfaces $\Sigma_{g,b}$ of genus $g$ with $b$ boundary components given by $$ \Gamma_{g,b} = \pi_0(Diff^+(\Sigma_{g,b}, \partial \Sigma_{g,b})) $$ satisfy homological stability in the sense that the homomorphisms of the homology groups obtained from gluing on various surfaces along the boundary components eventually become isomorphisms. For example, gluing on a pair of pants for $b \geq 1$ gives a homomorphism $$ \Gamma_{g,b} \to \Gamma_{g,b+1} $$ which induces an isomorphism in homology $H_r(\Gamma_{g,b}) \to H_r(\Gamma_{g,b+1})$ in the range $r > 1$, $g \geq 3r-2$.

These results were later generalised by Madsen, Weiss, Galatius, Randal-Williams, Hatcher, Vogtmann, Wahl and many others. It has influenced the way to think about cobordism categories and has revealed a lot about the structure of moduli spaces of surfaces (possibly with tangential structures).

This answer is much too short to give a huge and fruitful subject the attention it deserves. But maybe some of the people working in this area (some of which are active on mathoverflow) can expand on it.

Answer 4

Haynes Miller's resolution of the Sullivan conjecture.

Adams' solution if the Hopf invariant one problem.

Answer 5

Almgren–Pitts min-max theory. It leads to the proof of Willmore conjecture by Marques and Neves.

Answer 6

Following up the answer by Paul Siegel, in addition to Donaldson's spectacular result described by Paul, there are the papers of Sacks-Uhlenbeck on harmonic immersions of $2$-spheres into Riemannian manifolds and subsequent papers by Uhlenbeck on the "bubbling" phenomena for self-dual Yang-Mills connections and Taubes on gluing "bubbles" onto a self-dual Yang-Mills. These are the key technical results used by Donaldson in his thesis, as cited by Paul. They themselves laid the groundwork for a tremendous amount of work in geometric analysis since then.

Here are only a few that I recall offhand:

1) Minimal hypersurfaces (Schoen-Simon-Yau, Anderson)

2) Einstein manifolds (Gao, Anderson-Cheeger)

3) Recent work of Naber and Valtorta on stationary Yang-Mills connections

4) Recent work of Sung Jin Oh showing that similar phenomenon exists for the hyperbolic Yang-Mills equations

Answer 7

Hitchin's Self-duality paper.

In this paper, Hitchin constructs a hyper-kähler structure on the moduli space of flat irreducible connections by identifying them with Higgs pairs (holomorphic bundle together with a holomorphic endomorphism valued 1-form).

Initially, the construction was done for compact Riemann surfaces and bundles of rank 2, but generalize to Kähler manifolds, to punctured surfaces (parabolic structures) and to higher rank respectively pricipal bundles.

The paper is not about proving one single theorem (he also introduces the integrable system nowadays know as the Hitchin system), I hope you can count this as an answer, as the subject of Higgs fields is still developing with many interesting connections to other subjects like mathematical physics or geometric langlands.

Answer 8

Work done by many in low-dimensional topology that laid the groundwork for Thurston's work and his conjecture, which was proved by Perelman. I don't want to list names, since I'm sure I'll miss some key people.