How to tackle the smooth Poincaré conjecture

Question

The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. Freedman showed that this holds if we replace "diffeomorphism" by "homeomorphism", so another viewpoint would be that $S^4$ has no exotic smooth structures.

Restated: If $X$ is a connected, closed smooth 4-manifold with $\pi_1X=1$ and $H_*X\cong H_*S^4$, then $X\approx_\text{diffeo}S^4$.

$\newcommand\pt{\mathrm{pt}}$This was told to me by Michael Hutchings, who motivationally remarked a way to solve it (Edit: not an original suggestion of his):
Find a symplectic structure on $X-\lbrace \pt\rbrace$ which is standard near the puncture-point.
Then we're done by "Recognition of $\mathbb{R}^4$": Let $(M,\omega)$ be a noncompact symplectic 4-manifold such that $H_\ast(M)\cong H_\ast (\pt)$. Suppose there exist compact sets $K_0\subset M$ and $K_1\subset\mathbb{R}^4$ and a symplectomorphism $\phi:(M-K_0,\omega)\to(\mathbb{R}^4-K_1,\omega_\text{std})$. Then $\phi$ extends to a symplectomorphism $(M,\omega)\to(\mathbb{R}^4,\omega_\text{std})$, after removing slightly bigger compact sets.

I am ignorant to the size of this wall (the conjecture) and the ability to make an indent in it. But the above idea is pretty cool, even if not anymore helpful than the original statement. The Poincaré Conjecture (in dimension 3) was solved using Hamilton's idea of Ricci flow. This leads me to ask: Is there another idea proposed to tackle this conjecture? Or a failed attempt?

Answer 1

In principle the Ricci flow (with surgery) could also be used to prove the smooth Poincaré conjecture in dimension $4$.

There are some major problems to be overcome in this approach (problems which did not arise in dimension $3$, such as the absence of Hamilton–Ivey pinching estimates, and the new "hole-punch" singularities, as opposed to the "neck-pinch") and I know that some people are indeed working on these issues.

If you are interested, there are quite a few papers on Ricci flow on $4$-manifolds, starting with the work of Hamilton (Four-manifolds with positive curvature operator and Four-manifolds with positive isotropic curvature) and more recent work of Chen–Zhu (Ricci flow with surgery on four-manifolds with positive isotropic curvature) and Chen–Tang–Zhu (Complete classification of compact four-manifolds with positive isotropic curvature) and many others.

Answer 2

It seems to me your question implicitly assumes that the 4d smooth Poincaré conjecture (S4PC) is true. But if you were to take a poll of experts I think you would find that most of them suspect it is false. (See also Ryan's comment to the original question.) There are many large families of potential counterexamples to the S4PC. These are smooth 4-manifolds which are homotopy equivalent (and hence homeomorphic) to the 4-sphere, but no one knows how to prove that they are diffeomorphic to the 4-sphere.

(Akbulut and Gompf have made recent progress in showing that some, but not nearly all, of these potential counterexamples are standard 4-spheres, so I think people are more open to the idea that the S4PC might be true than they were four years ago.)

There are similarly many suspected counterexamples to the closely related Andrews–Curtis conjecture. Andrew Casson has some interesting (and unpublished) numerical evidence that the AC conjecture is false.

So if you are interested in this problem, you might want to spend as much time trying to prove that one of the potential counterexamples is an actual counterexample as you do trying to prove the conjecture.

Answer 3

The conjecture in question can also be thought of as the $4$-dimensional PL Poincare conjecture (because low-dimensional PL manifolds, including those of dimension $4$, carry a unique smooth structure) and this is how it is understood in most references mentioned below.

Some interesting approaches to the conjecture and its special cases can be found in several papers by Frank Quinn (some based on TQFTs and others in more classical spirit) and in some papers by Robert Craggs.

Much of the effort has been focused on the group-theoretic Andrews–Curtis conjecture, whose validity would imply that PL (or smooth) homotopy $4$-spheres given as handlebodies without $3$-handles are PL (or smoothly) standard. The latter assertion would also follow from the "Generalized Property R" conjecture. Then there's a separate industry of finding handlebody presentations of simply-connected $4$-manifolds without $3$-handles (see Problems 4.18 and 4.73 in Kirby's list, Section 6 of Funar and Gadgil - On the geometric simple connectivity of open manifolds, Gadgil's preprint One-handles and concordance kernels for smooth 4-manifolds and Corollary 3.2 of Quinn - Dual decomposition of 4-manifolds; note also Rasmussen's withdrawn paper Perfect Morse functions and exotic S^2 x S^2's (arxiv.org/abs/1005.4674)).

As observed by Curtis in an earlier paper "Homotopy manifolds" (in "Topology of 3-manifolds and related topics"), every compact contractible $2$-polyhedron PL embeds in some PL homotopy $4$-sphere; so if you find one that doesn't PL embed in $S^4$, you're done with the 4D PL Poincaré conjecture. This line of attack inspired some literature on PL embeddings of acyclic $2$-polyhedra in $S^4$ starting I guess with Zeeman's dunce hat paper; see Livingston - Observations on Lickorish knotting of contractible 4-manifolds for additional references.

There are numerous other approaches and related techniques, e.g. "Gluck twists" and "Akbulut corks". Kirby's problem list is a good source of further references prior to mid-90s; some other basic references on the Andrews–Curtis conjecture are collected here under (O1).

Answer 4

Fastforward $11+\frac12$ years, I thought I'd mention that I tried to do what my original post suggested, which is when I started my PhD and then when I completed my thesis it naturally spawned this project (not a shocker since Hutchings became my advisor). Use certain differential 2-forms (called "near-symplectic") to probe the geometry of potentially exotic 4-spheres — it led to a (null)result,

• Chris Gerig, No homotopy 4-sphere invariants using ECH = SWF, Algebr. Geom. Topol. 21(5): 2543-2569 (2021). DOI:10.2140/agt.2021.21.2543, arXiv:1905.10938, Project Euclid.

I do strongly believe there is some progress along these lines that can be made by being more clever, especially since gauge theory and symplectic geometry have been good at realizing/detecting exotic structures on smooth 4-manifolds.

Concerning the community's consensus on the truth/falsity of the conjecture, here is an anecdote (first-hand as I was the organizer): During one of the Harvard gauge theory seminars Cliff Taubes commented "I read on Wikipedia that the experts think the smooth Poincaré conjecture is true", and then everyone in the audience laughed; indeed most of the experts were in that audience that day.