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I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology here on MO, and it has been hard for me to find recent references in the field.

I do not mean to offend anyone who works in the field with this, I do love what I’ve seen of the field a lot in fact. But I am a little concerned about this. Any feedback would be appreciated, thanks!

Wojowu
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James Baxter
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    I guess it depends on what do you mean with differential topology. I have the impression that several subareas are very active (e.g. surgery theory) but I'm not very familiar with the field as a whole – Denis Nardin May 30 '19 at 11:58
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    "Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field? – Francesco Polizzi May 30 '19 at 12:04
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    I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford. – Todd Trimble May 30 '19 at 12:04
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    Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps. – Mark Grant May 30 '19 at 12:17
  • Ah, that does make a lot of sense @MarkGrant – James Baxter May 30 '19 at 12:18
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    @FrancescoPolizzi from my limited perspective, Donaldson theory is indeed not a dying field but a dead field. SW have put it out of business, I think. Or are there some interesting topological questions in which it is easier to use the Donaldson invariants? –  May 30 '19 at 12:55
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    @FrancescoPolizzi also I do not believe that symplectic topology is a part of differential topology. A question of definitions, you could say, but to me "differential" means continuously differentiable, smooth or real-analytic, definitely not symplectic. –  May 30 '19 at 12:56
  • @kartop_man I don't really understand your last comment. A symplectic manifold is smooth, and a symplectic form is a differential form... Yet you use the words as if they were opposite. – Najib Idrissi May 30 '19 at 13:05
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    @NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something? –  May 30 '19 at 13:15
  • @MarkGrant That looks like an answer to me. :) – Neal May 30 '19 at 13:43
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    @kartop_man If you prove a topological result in the category of Riemmanian Manifold that is righteously a Differential Topology result. Symplectic topology and Contact topology, are specific (big) fields of Differential Topology that earned their own name. Btw Donaldson theory is not as dead as you depict it, there is still people working on instantons on manifolds with special holonomy or in higher dimension or on extensions and applications of Instanton Floer Homologies and I believe there is more. – Overflowian May 30 '19 at 13:54
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    @WarlockofFiretopMountain OK, the criticism regarding the deadness of Donaldson's theory I accept. The first sentence I do not completely agree with, one should define what is "a topological result" first. There are for example symplectomorphisms which are smoothly isotopic to the identity but are not symplectically isotopic to the identity. Is this "a topological result"? Probably. I do not think that it is a differential topological result though. Either way, I believe that this is not a mathematical discussion, but a linguistic one, so we can just call each other wrong and stop there. –  May 30 '19 at 13:59
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    +1 "we can just call each other wrong and stop there" – Nik Weaver May 30 '19 at 14:36
  • @kartop_man I do not appreciate you putting words in my mouth. At what point did I claim that symplectic geometry was about smooth isotopies? What do you think differential topology is? – Najib Idrissi May 30 '19 at 17:43
  • @NajibIdrissi maybe I am misunderstanding you. What you did say is that "a symplectic form is a differential form". I inferred from this that you think that symplectic topology is about studying these symplectic forms up to the natural equivalence relation on random differential forms (which is smooth isotopy). What I said is that most of the interesting phenomena in symplectic topology are not about this (though some are). –  May 30 '19 at 17:54
  • Also, could you please explain to me whether Riemannian geometry is also to be considered as differential topology or not? Riemannian manifolds are smooth manifolds, and Riemannian metrics are tensors. –  May 30 '19 at 17:56
  • I think that differential topology is about studying the discrete invariants of smooth structures on topological manifolds. Symplectic topology is not necessarily a part of differential topology according to this definition (although there are some connections, something like the work of Abouzaid exploring how smooth structures on a manifold are reflected in symplectic structures on the cotangent bundle). This definition is probably not the only reasonable one, but I do believe that it is reasonable too. –  May 30 '19 at 18:02
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    @kartop_man Then you inferred wrong. What is natural to you is not natural to everyone. Riemannian geometry is closer to differential geometry (and yes, the name is usually symplectic geometry, and yes, if you ask people who work in the field, many will tell you that they prefer the name symplectic topology). – Najib Idrissi May 30 '19 at 18:03
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    @NajibIdrissi OK, I apologize for my wrong inference then (which resulted from me applying my definition of differential topology to your statement). You apparently have a different definition of differential topology, which I think is also reasonable, and your statement is compatible with that definition. –  May 30 '19 at 18:06
  • I am also inferring that you accused me of using the words wrongly before I accused you of that (essentially you applied your definition of differential topology to my statement, which indeed does not make much sense given your definition). –  May 30 '19 at 18:08
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    Questions about whether a field is dying or dead seem too opinion-based for MO. Compare https://mathoverflow.net/questions/332281/is-the-field-of-q-series-dead Voting to close. – Timothy Chow May 30 '19 at 19:03
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    @TimothyChow, are questions about dying fields dying? – LSpice May 30 '19 at 19:20
  • @NajibIdrissi so Najib Idrissi do you agree that you accused me of using words wrongly before I accused you of that or not? A definition that is natural for you is not natural for everyone. –  Jun 01 '19 at 20:26

2 Answers2

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I don't think differential topology is a dying field.

I'll interpret this as the classification of smooth manifolds and, more broadly, maps between them (immersions, embeddings, diffeomorphism groups). Also, I'll restrict to the finite-dimensional case.

There are related topics which are very active, usually studying smooth manifolds with extra structure, e.g. exterior differential systems, foliations and contact structures, symplectic and Riemannian geometry. I won't comment much on these areas.

The classification of smooth manifolds was quite successful in the 60s with the h- and s-cobordism theorems framing many classification problems in terms of surgery problems. The classification of exotic spheres was more-or-less reduced to problems in homotopy theory, the stable homotopy groups and Kervaire invariant problems. The study of these invariants is still active, but the techniques are more algebraic. Moreover, there is still an industry of studying Riemannian metrics on exotic spheres.

Maybe one of the biggest open problems now in differential topology is the cobordism hypothesis, originally formulated by Baez-Dolan, but reformulated by Lurie. This is formulated as a classification of "fully extended topological field theories" in terms of $(\infty,n)$-categories. His sketch of proof is regarded as incomplete, and a few groups are trying to fill in the details. From discussions I've had with experts, a big issue here is foundational results in differential topology. Lurie's outline relies on results about manifolds with corners, and I think that Schommer-Pries has filled in some details, but I think that the proof of the cobordism hypothesis is still incomplete.

Another (very special) problem that has received some attention is the Hirzebruch Prize Question:

Does there exist a 24-dimensional compact, orientable, differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, \hat{A}(X) = 1$, and $\hat{A}(X, T_C) = 0$?

Here $\hat{A}$ is the A-hat genus. The twisted Witten genus is supposed to be related to certain modular functions (McKay-Thompson series) associated with Monstrous Moonshine. I believe that Hopkins proved that a manifold with the right properties exists, but only in the topological category, and without the action of the Monster group. Daniel Allcock is working on constructing this manifold.

Shmuel Weinberger has championed the study of decidability questions in differential topology.

The Novikov conjectures would imply that $\mathcal{L}$-classes (certain combinations of Pontryagin classes) are invariant under homotopy equivalence of smooth aspherical manifolds. See a recent survey.

There is still active study of diffeomorphism groups. An active topic here is the study of homological stability for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.

Ian Agol
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    Here is Hopkins' 2002 ICM address where he gives a manifold solving Hirzebruch's prize question (without the action of the Monster group): https://arxiv.org/abs/math/0212397. The statement is the last sentence on page 303. It seems to me that the manifold is actually smooth, but I am probably missing something. – Aleksandar Milivojević Jun 01 '19 at 18:40
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    @AleksandarMilivojevic thanks, yes I agree it is smooth - when I answered the question, I couldn’t find a reference. – Ian Agol Jun 04 '19 at 15:24
  • Kuper's paper on the diffeomorphism groups is very nice. Unfortunately the chapter I was most interested in is just a tad brief--Ch. 24: The results of Kervaire-Milnor. Any recommendations on that? – Tom Copeland Sep 24 '20 at 18:33
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Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.

Mark Grant
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