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The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare 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 $X\approx_\text{diffeo}S^4$.

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 Poincare 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?

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as your formulation non-smooth is somewhat misleading let me clarify this point: Perelman used the Ricci flow (with surgery) to prove the 3-dimensional Poincare conjecture. In dimension three it doesn't matter whether you say homeomorphic or diffeomorphic. –  Robert Haslhofer May 20 '12 at 6:44
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On the other hand, there's an emerging point of view where people think it's likely that all compact 4-manifolds should admit countably-infinite many distinct smooth structures. This isn't a deep rationale -- it's just that for "sufficiently large" 4-manifolds, there appears to be always arguments to this effect. And "sufficiently large" manifolds have been getting smaller and smaller recently. See Stern's notes from this talk: math.uci.edu/~rstern/Cornell_2012.pdf –  Ryan Budney May 20 '12 at 7:38
    
Ah I was at this conference! Interesting point. –  Chris Gerig May 20 '12 at 8:58
    
One failed attempted was discussed in Ryan's question mathoverflow.net/questions/71031 –  Sergey Melikhov May 22 '12 at 8:00
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The business about finding a symplectic structure on $X-\{pt\}$ is not my original suggestion, nor a program that I am suggesting anyone try to carry out. It is just a motivational remark I made when introducting Gromov's theorem on the recognition of ${\mathbb R}^4$. –  Michael Hutchings May 22 '12 at 17:40
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3 Answers

up vote 13 down vote accepted

In principle the Ricci flow (with surgery) could also be used to prove the smooth Poincare' 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 (here and here) and more recent work of Chen-Zhu (here and here) and many others.

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Could you say why in principle (supposing one can get to grips with the singularities) the Ricci flow might help prove S4PC? I can imagine it might help produce an Einstein metric, but what then? –  Tim Perutz May 21 '12 at 16:10
    
Roughly the idea is the following. One would need to prove that a Ricci flow with surgery exists in $4D$ (analogously to the case of $3D$), and that on a topological $S^4$ the flow becomes extinct in finite time. Then one would have to prove a "canonical neighborhood" theorem which should imply that before becoming extinct, all components of the manifold have positive curvature operator (or locally they split as products of lower dim. manifolds with this property). A theorem of Hamilton (and Bohm-Wilking in any dimension) then implies that all such components are diffeomorphic to space forms –  YangMills May 22 '12 at 10:22
    
(or products of lower dimensional space forms). This program, initiated by Hamilton, has been carried out by Chen-Zhu if the initial metric has positive isotropic curvature, see arxiv:0504478. I think the general case is still pretty much open, and as the other comments suggest there is really no strong evidence for S4PC, so it's possible that this program cannot be carried out in general. –  YangMills May 22 '12 at 10:22
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There's several serious issues with this approach. There are $\kappa$-noncollapsed solitons in 4 dimensions, such as the 3D Bryant soliton x $\mathbb{R}$ or $S^1$, or $S^2\times \mathbb{R}^2$ which might occur as blow-up limits of finite-time singularities (the $S^3\times \mathbb{R}$ soliton or 4D Bryant soliton shouldn't be a problem). Doing surgery to proceed along the flow past these singularities might drastically change the topology of the manifold, and it's not clear that one can proceed without changing the topology while simplifying the geometry. –  Ian Agol May 22 '12 at 15:06
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Even if one can resolve the formation of singularities issue, it's also not clear how to prove that a flow on the 4-sphere might have a finite-time singularity. The proof in 3-dimensions by Colding-Minicozzi uses some special properties of minimal surfaces and their stability operators. It's not clear how to generalize this to 4-dimensions (although I certainly haven't thought about it). But there might be some other approach using e.g. Gauss-Bonnet. Finally, if one could prove that it converges to an Einstein metric or a soliton, then there still remains the classification of these! –  Ian Agol May 22 '12 at 15:13
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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.

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Is there a URL to Casson's numerical work? I'd done some myself (bit.ly/JzzsKu), but the exponential growth of the AC derivation tree "overpowered" the cluster I was working on. I'd be curious as to Casson's approach. –  Kelly Davis May 22 '12 at 15:56
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I heard about Casson's results from some lectures Casson gave at the 2003 Arkansas Spring Lecture Series. This page -- msp.warwick.ac.uk/gtm/2004/07 -- refers to this conference. Perhaps that could be a starting point of a web search. Yo'av Rieck was the main organizer, so perhaps he could help you track down some notes from Casson's talks. –  Kevin Walker May 22 '12 at 19:25
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If two manifolds are shown by Gompf to be diffeomorphic by Kirby calculus, then they are said to be Gompfomorphic. –  Ian Agol May 22 '12 at 23:29
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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 here, Gadgil's preprint and Quinn's Corollary 3.2; note also Rasmussen's withdrawn paper arxiv.org/abs/1005.4674).

As observed by Curtis in an earlier paper (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 Poincare 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 this review 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).

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