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I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. I'd really want to know the state of the question, since I'm self-studying the material for pleasure and I don't have anyone to talk about it. Please feel free to close this post if you think the topic is not appropriate for this site.

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological and smooth manifolds are widely studied and there are tons of books about them, PL topology seems to be much less popular nowadays. Moreover, I saw in some place the assertion that PL topology is nowadays not nearly as useful as it used to be to study topological and smooth manifolds, due to new techniques developed in those categories, but I haven't seen it carefully explained.

My first question is: is this feeling about PL topology correct? If it is so, why is this? (If it is because of new techniques, I would like to know what these techniques are.)

My second question is: if I'm primarily interested in topological and smooth manifolds, is it worth to learn PL topology?

Also I would like to know some important open problems in the area, in what problems are mathematicians working in this field nowadays (if it is still an active field of research), and some recommended references (textbooks) for a begginer. I've seen that the most cited books on the area are from the '60's or 70's. Is there any more modern textbook on the subject?

Thanks in advance.

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math.stackexchange.com/questions/70634/… addresses some of these questions. –  Daniel Moskovich May 5 '12 at 15:06
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I like the unnumbered questions in the end, but otherwise the question looks somewhat rhetorical and seems to call for a heated debate. If I'm primarily interested in programming, is it worth to learn mathematics? I heard that math is not nearly as useful as it used to be in computer science, due to new techniques developed in that subject. Pathetic, isn't it? And those books cited by mathematicians, some of them are so old! –  Sergey Melikhov May 5 '12 at 16:30
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@Daniel: Thanks very much! @Sergei: I get your point, but I think that it's not the same case as your analogy. Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today? And concerning the books, we all know that subjects in mathematics change, and some great textbooks in the past are not well suited to the present status of the area, because change of emphasis or discovery of new tehniques that make life easier. So I'm asking about "newer" books to know if there are references more suited to present PL topology. –  Carlos Sáez May 5 '12 at 16:47
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Even if you only care about smooth manifolds, I think it's worth having some familiarity with the language and basic ideas: some important isotopy/embedding theorems (e.g. Hudson) have written proofs in the literature only for PL manifolds but also hold in the smooth case. If you want to tweak these proofs maybe it's useful to speak the language. –  Jonny Evans May 5 '12 at 21:15
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PL topology is nowadays old fashioned because of its difficulty, as it often happens in math. Nevertheless, it's not uncommon that after decades a smart guy comes with new striking discoveries and gets its back to the mainstream. I hope this happens to PL topology! –  Fernando Muro Nov 7 '12 at 7:00
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6 Answers

Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today?

According to a recent poll by the Central Planning Commitee for Universal Education Standards, some geometric topologists don't have a clue about regular neighborhoods, while others haven't heard of multijet transversality; but they all tend to be equally excited when it comes to Hilbert cube manifolds.

some recommended references (textbooks) for a beginner

Rourke-Sanderson, Zeeman, Stallings, Hudson,

L. C. Glaser, Geometrical combinatorial topology (2 volumes)

Is there any more modern textbook on the subject?

Not really (as far as I know), but some more recent books related to PL topology include:

Turaev, Quantum invariants of knots and 3-manifolds (chapters on the shadow world)

Kozlov, Combinatorial algebraic topology (chapters on discrete Morse theory, lexicographic shellability, etc.)

Matveev, Algorithmic topology and classification of 3-manifolds

2D homotopy and combinatorial group theory

Daverman-Venema, Embeddings in manifolds (about a third of the book is on PL embedding theory)

Benedetti-Petronio, Branched standard spines of 3-manifolds

Buchstaber-Panov, Torus actions and their applications in topology and combinatorics

Buoncristiano, Rourke, and Sanderson, A geometric approach to homology theory (includes the PL transversality theorem)

The Hauptvermutung book

Buoncristiano, Fragments of geometric topology from the sixties

Also I would like to know some important open problems in the area, in what problems are mathematicians working in this field nowadays

I'll mention two problems.

1) Alexander's 80-year old problem of whether any two triangulations of a polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or consructible ball), common subdivisions exist. See also remarks on the Alexander problem by Lickorish and by Mnev, including the story of how this problem was thought to have been solved via algebraic geometry in the 90s.

2) MacPherson's program to develop a purely combinatorial approach to smooth manifold topology, as attempted by Biss and refuted by Mnev.

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Thanks for the answer, specially the great list of references. –  Carlos Sáez May 5 '12 at 21:31
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I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a foundational subject to most mathematicians. A large proportion of low dimensional topologists use the foundational theorems in PL topology as black boxes, certainly without understanding or having read the proofs, and in fact they can do good mathematics that way. In the smooth category, the situation is even worse- I'm sure that there are very few people in the world who understand the proof of Kirby's Theorem, which is a difficult result, but it gets used all over low dimensional topology as a black box. Indeed, the fact that a diffeomorphism of $S^2$ extends to the $3$--ball is fundamental, under the hood everywhere, and highly non-trivial.

So you can be a manufacturer, or you can be a consumer. As a consumer, maybe you don't need to know PL topology beyond the basics that you need in order to understand simplicial homology and other basic constructions. A more sophisticated consumer might need more- I don't for example know a concrete smooth construction of linking pairings (the PL construction is in Schubert)- and in general, cell complexes allow you to work explicitly and concretely. PL proofs, if you read and care about proofs of fundamental results, tend to be shorter and easier than smooth proofs, which is not surprising because a-priori there is so much less structure which has to be carried around. This was indeed why Poincaré first considered triangulated manifolds; because of the technical facility which they afforded him. As a counter-point, I should point out Smale's comment in the introduction to in 1963 paper A survey of some recent developments in differential topology (which I recommend that you read, as it discusses your question):

It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet...

Another aspect, which is not to be sneezed at in today's world, is that PL manifolds are better suited to computers. This is indeed the focus of Matveev's book on "algorithmic topology".

Finally, as a PL question, I nominate:

Open problem: Construct a discrete $3$-dimensional Chern-Simons theory, compatible with gauge symmetry, replacing the path integrals of the smooth picture (which are not mathematically well-defined) with finite dimensional integrals.
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We should ask Smale how he would prove, at the time of writing the quote, that fibers of generic smooth maps are homotopy equivalent to CW-complexes. The only proof known to the MO community (mathoverflow.net/questions/94404) is based on the (trivial!) combinatorial counterpart of this statement, and didn't appear until Thom's conjecture on triangulation of smooth maps was proved by Andrei Verona in 1984. –  Sergey Melikhov May 8 '12 at 2:50
    
@Sergey: That's a good point, but I think that it's fair to suppose that he would not have considered that to be one of the main theorems of differential topology. Indeed, I would argue that cell complexes are part of the PL world, and that the smooth world is more about handles. –  Daniel Moskovich May 8 '12 at 13:51
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Daniel: then let's replace "are homotopy equivalent to CW-complexes" by "have finitely generated cohomology". (The domain and the range are closed smooth manifolds.) –  Sergey Melikhov May 8 '12 at 22:48
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As a counterpoint to the Open Problem at the end, since Poincare himself many folks have tried to find a combinatorial proof that every closed simply connected 3-manifold is homeomorphic to the 3-sphere. They may still be trying... –  Lee Mosher May 9 '12 at 4:06
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Some points I didnt see mentioned above: the basic results of geometric topology: tubular neighborhood theorem, transversality, xetc. have easy smooth proofs, somewhat technical PL proofs, and difficult (Kirby-Siebenmann+surgery theory) TOP proofs. Historically TOP came after the development of Smooth and PL, but in the end, the formalism in high dimensions was entirely encoded in the algebraic topology of the classifying spaces $B$Diff$=B$O, $B$PL$, B$TOP. The bottom line is that many high dimensional problems can be "reduced" to algebraic topology of these classifying spaces, and so it isn't that PL isn't interesting, just that it can be treated (say in surgery theory, or smoothing theory) on equal footing with the other two, as a black box, without really knowing anything specific about the nuts and bolts of PL topology (just as you can understand most smooth topology without knowing a careful proof of the implicit function theorem).

Following the success of high dimensional topology, the focus in geometric topology shifted to low dimensions starting in the early 1980s, and as Dylan comments there is no difference between PL and Diff in low dimensions, so that the more familiar smooth methods suffice, and more recently trained topologists have no reason to study PL methods if their focus is on low dimensions.

As a topology student, it is probably good for you to have some familiarity with the surgery exact sequence, $$\mathcal{S}_{PL}(X)\to [X,G/PL]\to L(\pi_1(X))$$ and its counterparts with PL replaced by Diff or TOP (i.e. what the objects and maps are in this sequence). Knowing the early big successes in your area will give you a better appreciation of what is happening in it now.

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Paul, your opinions on easy and hard proofs of the "etc." results, and on the needs of younger topologists, are perfect examples of the long expected heated debate... I'll leave it there, but how about the presumably less controversial issue of whether basic results of smooth topology can be proved at all (easy or hard) without using PL topology? In particular, that fibers of generic smooth maps between smooth manifolds are "homotopy equivalent to CW-complexes"? (This was a recent MO question, mathoverflow.net/questions/94404) –  Sergey Melikhov May 7 '12 at 2:32
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Even the non-"etc." examples are not so plain. The PL analogue of Sard's theorem is certainly easier. It says that if you take any point $p$ in the interior $U$ of a top-dimensional simplex in the range of a simplicial map $f$, then $f^{-1}(U)$ is PL homeomorphic to $U\times f^{-1}(p)$, and this is trivial to prove (as opposed to the full PL transversality). The existence of a regular neighborhood is a tautology (with definitions as in the Rourke-Sanderson book), but that of a tubular neighborhood needs proof; like other smooth proofs and unlike PL proofs, it depends on years of Calculus... –  Sergey Melikhov May 7 '12 at 3:07
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Let us not forget that smooth codimension $k$ embeddings have normal bundles with structure group $\text{O}(k)$, whereas this analogy does not hold in the PL-case (one has to use block bundles instead). –  John Klein May 7 '12 at 17:52
    
John: there is a view (expressed already in the Rourke-Sanderson Annals paper series) that it is block bundles that are the right notion of a bundle in the PL category. For instance I wonder how you would do something beyond definitions with PL bundles, like Euler or Stiefel-Whitney classes or the umkehr map, without using the theory of block bundles. For block bundles these things are done in the Bouncrisiano-Rourke-Sandrson book. –  Sergey Melikhov May 8 '12 at 1:40
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Sergey: if one wants to understand automorphisms of PL manifolds, one cannot dispense with $PL(k)$. On the other hand, there's a sense in what you've claimed: notice $O(k+1)/O(k) = S^k$, and as $k$ varies, this gives the sphere spectrum (this is responsible for our understanding of the the Euler class). However, the spectrum associated with $PL(k+1)/PL(k)$ is very complicated: it's Waldhausen's $A(∗)$! On the other hand, the spectrum associated with $\widetilde{PL}(k+1)/\widetilde{PL}(k)$ is much easier: by Haefliger it's the sphere spectrum. –  John Klein May 9 '12 at 13:20
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Disclaimer: What follows is probably a bit off-topic for this site, but no more than the original questions, numbered one and two. In fact I suspect that this answer attempts to address just what the OP really wanted to ask ("isn't PL topology useless?") by posting those two lightly euphemistic questions. If there was an active meta thread for closing this question, I'd rather put this answer there.

Some topologists, perhaps the majority, tend to think that smooth and topological manifolds are "present in nature" and are the genuine objects of study in geometric topology, while PL topology is a somewhat artificial, unnatural construct, and matters just as long as it is helpful for the "real" topology. I've heard this opinion stated explicitly once, and I see a lot of this kind of attitude in this thread. In fact I think this philosophy/intuition is sufficiently familiar to nearly everyone that I don't need to elaborate on it. Moreover, I suspect that a lot of people are not even aware that it is not the only possible religion for a topologist, or else they would be more considerate to the heretics in stating their strong opinions.

I'd like to discuss one other philosophy/intuition then, according to which both smooth and topological manifolds are obviously artificial, highly deficient models for what could be "present in nature", whereas the PL world is much "closer to the reality". I don't consider myself a practitioner of this or any other religion; what follows should be regarded as said by a fictional character, not by the author.

1) As is well-known, the predisposition to seeing continuous and smooth as more natural than discrete is historical, following centuries of preoccupation with derivatives and (later) limits. Quantum physics and computer science may be changing the tide, but they don't usually compete with Calculus in a mathematician's education, at least not in the initial years.

Here is a simple test. When you fold a sheet of paper, what is the intuitive model in your imagination: is it a smooth surface (when you look with a loupe at the fold), a cusp-like singularity (generic smooth singularity), or an an angle-like singularity (PL singularity)? No matter what is your subconscious preference, I bet you didn't base it on considerations of individual photons detected by the eye. But you could have based it on your previous experience with abstract models of surfaces, which is not independent of the historically biased education. (Just for fun, I wonder if your intuitive model would change if the paper sheet is folded second time so as to make a corner - which is unstable as a singularity of a smooth map $\Bbb R^2\to\Bbb R^2$, but has a stable singularity in the link.)

2) On a molecular scale, the sheet of paper of course doesn't fit the model of a smooth surface, and although it is arguably not "discrete" or "PL" on a subatomic scale, the smooth surface model isn't restored either. Similarly, as is well-known, Maxwell equations and general relativity (which I guess are among the best reasons to study smooth topology) don't work at very small scales. The problem is that this "imperfection" of matter doesn't usually shake one's belief in "perfect" physical space. But it is perfectly consistent with modern physics (for those who don't know) that physical space is kind of discrete at a sub-Planck scale, as in loop quantum gravity (which is somewhat reminiscent of PL topology!). It is also consistent with the present day knowledge, and indeed derivable in variants of the competing string theory, that a finite volume of physical space can only contain a finite amount of information, as with the holographic principle. (In fact I didn't see much discussion of possible alternatives to this principle, many physicists appear to take it for granted.) I'm getting on a slippery slope, but finite information does not sound like it could be compatible with limits that occur in derivatives (which returns us to MacPherson's program on combinatorial differential manifolds) and especially with Casson handles that occur in topological manifolds.

The fictional character is now saying that his religion teaches him to avoid concepts based on inherently infinitary constructions, because they are likely to be unnatural, in the sense of the physical nature which might simply have no room for them (and even the question of whether it does is not obviously meaningful!). Ironically, this is quite in line with Poincare's philosophical writings, where he argued at length that the principle of mathematical induction is not an empirical fact.

3) The fictional character goes on to say that this is not just the crazy metaphysics that displays the warning, but also Grothendieck with his "tame topology" which inspired a whole area in logic (initiated by van den Dries' book Tame topology and o-minimal structures). Here is a short quote from Grothendieck:

It is this [inertia of mind] which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom "wildness" is a fatal necessity, rooted in the nature of things.

My approach toward possible foundations for a tame topology has been an axiomatic one. Rather than declaring [what] the desired “tame spaces” are ... I preferred to work on extracting which exactly, among the geometrical properties of the semianalytic sets in a space $\Bbb R^n$, make it possible to use these as local "models" for a notion of "tame space" (here semianalytic), and what (hopefully!) makes this notion flexible enough to use it effectively as the fundamental notion for a “tame topology” which would express with ease the topological intuition of shapes.

Grothendieck dismisses from the start PL and smooth topology as possible forms of tame topology, because

(i) they're "not stable under the most obvious topological operations, such as contraction-glueing operations", and

(ii) they're not closed under constructions such as mapping spaces, "which oblige one to leave the paradise of finite dimensional spaces".

I'm not familiar with "contraction-glueing operations", nor is Google. Perhaps someone fluent in French could explain what (i) is supposed to mean? My first guess would be that this could refer to mapping cylinder, mapping cone or other forms of homotopy colimit, but PL topology is closed under those (finite homotopy colimits).

Edit: Indeed, it is clear from the preceding pages that by "gluing" Grothendieck means the adjunction space, which he also calls "amalgamated sum". In particular, he says:

It was also clear that the contexts of the most rigid structures which existed then, such as the "piece-wise linear" context were equally inadequate – one common disadvantage consisting in the fact that they do not make it possible, given a pair $(U,S)$ of a "space" $U$ and a closed subspace $S$, and a glueing map $f: S\to T$, to build the corresponding amalgamated sum.

There is, of course, no problem with forming adjunction spaces in the PL context. Perhaps Grothendieck was just not aware of pseudo-radial projection or something. End of edit

As to (ii), there now exists some kind of an infinite-dimensional extension of PL topology, which includes mapping spaces and infinite homotopy colimits up to homotopy equivalence (and hopefully up to uniform homotopy equivalence, which would be more appropriate in that setup). Besides, there are, of course, Kan sets, which are closed under Hom, but they arguably don't belong to tame topology in any reasonable sense because they quickly get uncountable (in every dimension, in particular, there are uncountably many vertices) and even of larger cardinality.

In any case, logicians, who tried to set up Grothendieck's aspiration in a rigorous framework of definability (see Wilkie's survey), do now have the "o-minimal tringulability and Hauptvermutung" theorem, saying roughly that tame topology (as they understood it) is the same as PL topology. Still more roughly (perhaps, too roughly) is could be restated as "topology without infinite constructions is the same as PL topology".

Even if smooth topology will some day be reformulated in purely combinatorial terms, it is highly unlikely that it can be characterized by purely logical constraints. From this viewpoint, smooth topology is primarily justified by its role in applied math and natural sciences, but is no less and no more fundamental than symplectic topology or topology of hyperbolic manifolds.

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One thing meant by "gluing", I suppose, is that extra information (collars) is needed to specify the result of gluing smooth manifolds along a boundary. But anyway, my fictional character argues for smooth topology: Only natural numbers are "real". Calculus is "real", because it provides a low algorithmic complexity setting to answer natural number problems. Smooth manifolds are real because they are spaces on which calculus can be performed. PL manifolds make sense as discrete models for smooth manifolds; or else you need to argue that they have "independent" existence. –  Daniel Moskovich May 9 '12 at 7:32
    
... (cont.) So there is a lot of interesting structure (geometry, flows, analytic structure...) which you can impose on a smooth manifold. A manifold is a world to live in: clearly, a lot of dynamics can take place in smooth manifolds. But there seems to be no life on a PL manifold- it's a barren, cubist wasteland. Thus quoth my fictional character. IRL, I don't know, and I'm happy that there are adherents of both "religions" (and others besides) amongst topologists. –  Daniel Moskovich May 9 '12 at 7:43
    
Daniel, thank you for feedback. (I'm still puzzled by Grothendieck's contraction-gluing; the result of gluing two PL manifolds along a PL homeomorphism of their boundaries doesn't need any extra information.) There are, of course, things like geometric structures on cell complexes (popular in geometric group theory, see e.g. the Bridson-Haefliger book), harmonic functions on simplicial complexes (see R.Forman's 1989 paper in Topology), combinatorial Gauss-Bonnet formula (see Yu Yan-Lin's 1983 paper in Topology), connections and parallel transport on PL manifolds (see M.A.Penna's 1978 paper... –  Sergey Melikhov May 9 '12 at 9:42
    
...in Pacific J Math, and also arxiv.org/abs/math/0604173) and of course PL de Rham theory (see D. Lemann's 1977 paper in Asterisque, R.G.Swan's 1975 paper in Topology, Bousfield-Gugenheim 1976 AMS memoir). Of course, discrete analysis is motivated by the smooth case (Smale should have been more specific!) so no wonder it lags behind. The problem with PL topology is I think that it has indeed been largely deserted since 1970s and as a consequence is now underdeveloped and barely taught to students. I'm not sure that there's any internal reason for that, it could be entirely cultural. –  Sergey Melikhov May 9 '12 at 10:05
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PL topology is popular in quantum topology where some invariants (e.g Turaev-Viro) are defined by fixing a triangulation and the checking invariance under some standard moves.

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It's worth commenting (for those that don't know) that PL topology is the same as smooth topology in low dimensions (up to 6). –  Dylan Thurston May 6 '12 at 2:57
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...which is a highly nontrivial fact (particularly Cerf's theorem, implying that smooth structures are unique on PL 4-manifolds and exist on PL 5-manifolds) better stated as "PL topology includes smooth topology in low dimensions" because PL topology is not just about PL manifolds but also about polyhedra (not to mention PL maps). Even that is not quite accurate, because families of low-dimensional smooth structures don't boil down to those of PL structures (see mathoverflow.net/questions/7892), so no wonder that Haefliger's smooth knots of $S^3$ in $S^6$ are trivial as PL knots. –  Sergey Melikhov May 8 '12 at 4:05
    
... Bringing in the morphisms, PL maps include (by another highly nontrivial result) generic smooth maps, as well as smooth maps that belong to generic families, and the inclusion is strict (for maps between manifolds) starting from very low dimensions (2 to 1). Arbitrary smooth maps easily have arbitrary compact metric spaces as point-inverses, so I'm not sure if they belong to smooth topology. For sure, mapping cylinders of (very low-dimensional) generic smooth maps aren't smooth manifolds, but one can't deny their place in PL topology. –  Sergey Melikhov May 8 '12 at 4:36
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On a smooth manifold we have Ricci flow. What is the analogue for a PL manifold?

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You should ask this as a separate question, rather than adding here. –  arsmath Nov 7 '12 at 13:10
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@arsmath: I had the same initial reaction until I realized that OP is also asking for a list of open problems in PL topology. Defining a combinatorial analogue of Ricci flow (say, in dimension 3) is a well-known open problem. If such flow exists, it could lead to a more constructive proof of, say, Poincar\'e conjecture. –  Misha Nov 7 '12 at 13:56
    
Indeed, for further related discussion see Bruce Westbury's own question mathoverflow.net/questions/65691/… –  j.c. Nov 7 '12 at 15:22
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It would certainly have helped if Bruce had added a link to his own question, and/or described a little context. –  S. Carnahan Nov 8 '12 at 8:19
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