Think of euclidean 3-space, as a linear subspace of euclidean 4-space, .  So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space.  And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram).   At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot. 

But there’s a more uniform version of this construction.  I think I first learned of it from Rolfsen’s textbook.  It sits most naturally in a slightly different formalism. 

is a smooth embedding and   whenever .

is called the space of long knots in .  There is a natural inclusion map .  I claim it is a null homotopic map. 

The idea is pretty simple. When you perform the inclusion, there is a new direction in orthogonal to the original . Let’s call the unit vector in that direction . The idea is to take a little bump function , and add to . Choose so that it is strictly increasing along the interval , and then have it decrease to zero quickly near .

There is a straight-line isotopy from to , and also from to where is the standard inclusion, i.e. always. Similarly there is a straight-line isotopy from to . Assemble these three maps together and you have your null-homotopy of .

Once you see this null-homotopy, notice that there actually two such null-homotopies! If instead of choosing to be increasing along , we could have chosen it to be decreasing along . If you assemble those two null-homotopies together, you get a map , or equivalently .

My question for people here is, is this map null-homotopic?

I don’t know the answer to this question. From the perspective of the Vassiliev spectral sequence, or the Goodwillie embedding calculus, this is a difficult map to understand. You can check that this map is zero on rational homology and rational homotopy groups. Not enough is known about the torsion in homotopy or homology to say what’s going on there — in a way that’s part of why I love this question. It’s also possible one of you will notice there’s a naive null-homotopy of this map. I just don’t see it.

Do any of you have a feeling for how knot concordance should be organized, say if one was looking for some global structure?    In the purely 3-dimensional world there are many very “tidy” ways to organize knots and links.  There’s the associated 3-manifold, geometrization.  There’s double branched covers and equivariant geometrization, arborescent knots and tangle decompositions.  I find these perspectives to be rather rich in insights and frequently they’re computable for reasonable-sized objects.  

But knot concordance as a field feels much more like the Vassiliev invariant perspective on knots: graded vector spaces of invariants.  Typically these vector spaces are very large and it’s difficult to compute anything beyond the simplest objects. 

My initial inclination is that if one is looking for elegant structure in knot concordance, perhaps it would be at the level of concordance categories.  But what kind of structure would you be looking for on these objects?   I don’t think I’ve seen much in the way of natural operations on slice discs or concordances in general, beyond Morse-theoretic cutting and pasting.   Have you? 

The Lins–Lins challenge

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post).  But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

(I’ve taken the liberty of copying the diagrams from their paper.)

The two manifolds are both hyperbolic, and calculations have been unable to distinguish their volumes or Witten–Reshetiken–Turaev invariants, up to many decimal places.  On the other hand, Lins–Lins’s techniques for finding homeomorphisms have also failed.  They (rather optimistically, in my opinion) bet that the failure of their techniques indicates that these manifolds really aren’t homeomorphic, and challenge the broader 3-manifold and combinatorial group theory communities to resolve the question.

They also note that, by Mostow Rigidity, we may equivalently try to determine whether or not the fundamental groups are isomorphic. They helpfully provide presentations.

(Again, I’ve copied this straight from their paper.)

There are some obvious things that one might attempt and which they don’t mention, such as computing the homology of the small-index subgroups of these groups.  Presumably they forgot to mention this, but it might be worth a try if anyone has any spare time.

(Added: This was a good idea! Nathan Dunfield did exactly this (see the comments below) and found that $G_1$ has a subgroup of index six with infinite abelianization, whereas $G_2$ does not. Readers are also referred to Nathan’s comments on the usefulness of SnapPea for this sort of problem.)

Regardless of the difficulty of this specific question, their broader complaint is valid.  Although we know that the homeomorphism problem for hyperbolic 3-manifolds (and, more generally, for all 3-manifolds) is solvable in theory by a remarkable algorithm of Sela, it has never been practical to even think about implementing that algorithm [S].

Aside: Sela’s algorithm

Sela’s algorithm is a tour de force; it solves the isomorphism problem for all ‘rigid’ torsion-free word-hyperbolic groups and, in fact, with rather more work, for all word-hyperbolic groups [DGu] and even all toral relatively hyperbolic groups [DGr].  But this very generality makes it daunting to think about implementing in practice.  One of the bounds is obtained from a classic Sela-style argument by contradiction on a limiting R-tree, so we get no control over this term. To add to the difficulty, the proof reduces the calculation to a set of equations and inequations over a non-abelian free group, which can be solved by Makanin’s algorithm [Mak]. Implementing Makanin’s algorithm is itself a highly non-trivial task.

The Scott–Short algorithm

I want to spend the rest of this post describing a remarkable recent observation of Scott–Short, which deserves to be much better known [SS].  They have a described a completely new solution to the homeomorphism problem for hyperbolic 3-manifolds, which relies solely on the geometry of hyperbolic 3-space. Their algorithm is technically much simpler than Sela’s, and huge credit must go to them for finding a better way of doing things.

Although I don’t for a moment think that their algorithm would be efficient as it stands, its simplicity and geometric nature bring the problem within reach.  Anyone with a working knowledge of hyperbolic geometry can understand it, which means that anyone can think about ways to improve it. Also, it has the huge advantage that it reduces the problem to calculations over the complex numbers,  rather than free groups.

Suppose we are given closed, orientable, hyperbolic manifolds $M_1$ and $M_2$ that we want to identify or distinguish. The starting point for Scott–Short is an algorithm of Manning (originally described by Casson) [Man], which constructs a fundamental domain $P_i$ in hyperbolic 3-space, together with side-pairing isometries $\{g_k^{(i)}\}$, for the manifold $M_i$. By Mostow Rigidity, we need to determine whether or not the subgroups

$G_i=\langle g_k^{(i)}\rangle \subseteq P⁢S⁢L_2⁢(\mathbb{C})$

are conjugate.

In practice, the fundamental domain and the side pairings are the sort of information that one usually already has about a given hyperbolic 3-manifold, so one shouldn’t need to implement Manning’s algorithm.  Perhaps an expert can tell me whether or not Snappea provides this information?  (It’s fine to go through non-exact computations like Snappea. If a candidate isomorphism is found, it’s easy to confirm whether or not it’s genuine; likewise, non-exact estimates are fine to compute the bounds we would need for a proof of non-isomorphism.)

Deciding whether or not two finite subsets of $PSL_2(\mathbb{C})$ are conjugate is just a matter of solving a system of equations over $\mathbb{C}$, which is relatively straightforward.  The difficulty arises because, even if $G_1$ and $G_2$ are isomorphic, the generating sets we have been given may be very different. Therefore, a priori, wenhave infinitely many different possible conjugacies to check.

Scott and Short use the geometry of the fundamental domains $P_i$ to reduce this to a finite number.  They define the n-star of $P_i$ to be the union of the translates of $P_i$ by the elements of word length at most n in $G_i$.  That is, in the tiling of hyperbolic 3-space by copies of $P_i$ it’s the set of tiles that are at most n steps away from $P_i$ itself.

They observe that, for any r, we can compute an n so that the metric ball of radius r around $P_1$ is contained in the n-star of $P_2$.  Since

$g_k^{(1)}⁢P_1\subseteq B⁢(P_1,\mathrm{diam}⁢P_1)$

for any k, it follows that we can compute an n so that the $g_k^{(1)}$ are all contained in the ball of diameter n in the word metric on $G_2$. We can then do the same thing the other way round, which reduces the problem to a finite computation as advertised, and we are done!

I recommend the Scott–Short paper, which is very readable.  It seems likely to me that an improvement of their algorithm, although it would no doubt be very slow on worst-case input, might well be effective enough to resolve questions like the Lins–Lins challenge. So I’ll finish with a challenge of my own.

Challenge:  Try to implement the Scott–Short algorithm!  Find ways to improve it!

Bibliography

[DGr] Dahmani, Groves, ‘The isomorphism problem for toral relatively hyperbolic groups’, Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 211–290.

[DGu] Dahmani, Guirardel, ‘The isomorphism problem for all hyperbolic groups’, Geom. Funct. Anal. 21 (2011), no. 2, 223–300.

[LL] Lins, Lins, ‘A challenge to 3-manifold topologists and group algebraists’, arXiv:1304.5964v1

[Mak] Makanin, ‘Decidability of universal and positive theories of a free group,’ (English translation) Mathematics of the USSR-Izvestiya 25 (1985), 75–88; original: Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 48 (1984), 735–749.

[Man] Manning,  ‘Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem’, Geom. Topol. 6 (2002), 1–25

[SS] Scott, Short, ‘The homeomorphism problem for closed 3-manifolds’, arXiv:1211.0264v1

[S] Sela, ‘The isomorphism problem for hyperbolic groups. I’, Ann. of Math. (2) 141 (1995), no. 2, 217–283.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is -Efficient triangulations and the index of a cusped hyperbolic -manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!).

As discussed previously on this blog here, there is an exciting new paradigm in M-theory, of a family of maximally symmetric conformal field theories is six dimensions which underlie lower dimensional supersymmetric field theories, inducing geometric descriptions for them. The archetype for such a concept is Witten’s TQFT for the Jones polynomial (a dimensional TQFT undelying the Jones polynomial), and Witten’s extension of it which gives a gauge theoretical description for Khovanov homology (see his`Fivebranes and knots’ paper).

The six dimensional theories are not understood mathematically, but they do lead to mathematical predictions for relationships between lower dimensional theories, that are mathematically better understood. In particular, Dimofte-Giaotto-Gukov show that a six dimensional theory implies a correspondence between an `index invariant’ of a triangulated cusped hyperbolic –manifold and a gauge theoretic invariant. The gauge theory is independent of the triangulation, and so you would expect from Physics that the index invariant ought to be a topological invariant of the -manifold. The reason that you should care is that the index invariant is an analytic continuation of the coloured Jones polynomial in some physics sense (which I don’t understand), yet it is intimately connected to the geometry of a cusped hyperbolic –manifold, in particular to its normal surfaces. I imagine that the point is that the coloured Jones polynomial corresponds to a perturbative expansion around the trivial flat connection, whereas the index invariant is associated to the whole gauge theory (and is thus non-perturbative).

The Volume Conjecture relates the asymptotic behaviour of coloured Jones polynomials with the hyperbolic volume; the topic of the `GHRS’ preprint is to relate the index invariant with normal surface theory of hyperbolic -manifolds, which in particular proves that the index invariant is a topological invariant which does not depend on the triangulation, within a class of triangulations depending only on the manifold (can this be strengthener further?). The idea would then be that it is a `more natural’ object than the coloured Jones polynomial, because it comes straight from gauge theory rather than from some perturbative expansion around some connection; and that its relationship with hyperbolic geometry would be easier to understand. I imagine that the volume conjecture might somehow follow from a parallel and easier conjecture about the index invariant, but I don’t know how.

Maybe the index invariant will be the next big thing?

What I like about this work is that it links quantum topology to normal surface theory, and to the combinatorics of triangulations of cusped hyperbolic -manifolds, in a compelling and explicit way (other connections between these fields have been investigated by others, but perhaps this way is more physically justified because the invariant is somehow non-perturbative). And I expect that this will be better understood by all following the Nha Trang conference.

This is big news!!

In the geometric approach, a knot is viewed as its complement- a 3-manifold with boundary, which you might imagine as what is left after a small worm eats out a knot-shaped path of that big apple that is . This complement can be equipped with a geometric structure (an instance of geometrization), and we can measure invariants such as its hyperbolic volume. You might fancy thinking of the geometric approach as a knot theorist’s “general relativity”.

In the quantum approach, on the other hand, a knot is viewed as its diagram, that is a configuration of crossings (minute, subatomic if you will, neighbourhoods in which “something happens”), and is understood by how subsets of those crossings connect together combinatorially. So you might think of the quantum approach as a knot theorist’s “quantum mechanics”.

Like general relativity and quantum mechanics in Physics, geometric and quantum topology of knots are both very useful, but the relationship between them remains a mystery. Do we get geometry if we zoom out of a knot diagram (I certainly hope so!). If so, how, and (more metaphysically), why?

There are a number of conjectures which relate geometric and quantum topology. Most famous are the Volume Conjecture, the AJ conjecture, and the Asymptotic Expansion Conjecture. The “story” for these conjectures seems to factor through physics, and they seem a bit technical and unenlightening to me- I’ve never succeeded in getting excited about any of these conjectures, despite having tried.

Fiedler’s approach, on the other hand, looks great! Quantum invariants can be viewed as combinatorial 0-cocycles in a moduli space of knots. What happens if we try to compute higher cocycles in the same (or a closely related) space? 1-cocycles, say? This idea would look natural to a smooth topologist or to a hard-core algebraic topologist, but it seems to have been off the radar of the quantum topological community- bravo Fiedler!

I defer to Fiedler’s introduction:

The connection to geometry is based on a result of Hatcher. It is well known that the classification problem for knots is equivalent to the classification problem of long knots, i.e. a smoothly embedded arcs in 3-space which go to infinity outside a compact set as a straight line. Let be a long knot and let be its topological moduli space. There are two natural loops in : the rotation of around the long axis, which we call and another loop, which we call Hatchers loop (compare [17]). It is defined as follows: one puts a pearl (i.e. a small 3-ball B) on the (closure of the) knot in the 3-sphere. The part of in is a long knot. Pushing B once along the knot induces Hatcher’s loop in . The following theorem is an immediate consequence of a result of Hatcher (see [17]).

Theorem 1 (Hatcher)
Let be a long knot which is not a satellite (i.e. there is no incompressible torus in its complement in the 3-sphere). Then and represent the same class in if and only if is a torus knot.

The hard direction of this theorem is of course to prove that for a hyperbolic knot these two loops are not homologous. This follows from the minimal models for the topological moduli spaces of hyperbolic and of torus knots which were constructed by Hatcher (see [8] and [7] for the case of satellites). Hatchers construction is mainly based on very deep results in 3-dimensional topology: the Smith conjecture, the Smale conjecture, the Linearization conjecture (which is a consequence of the spherical case in Perelmann’s work) and the result of Gordon-Luecke that a knot in the 3-sphere is determined by its complement.

In this paper we construct a combinatorial 1-cocycle for which is based on the HOMFLYPT invariant, see Theorem 4 in Section 11. It is called R(1) (“R” stands for Reidemeister).

Fiedler then reproves Hatcher’s Theorem, for the figure eight knot, using only quantum topology. This is a major triumph! Quantum proofs of geometric-looking facts like that are rare and precious. It indicates to me that there is real gold in Fiedler’s approach. The conjecture is that the the crux of the proof works for any knot, leading to a new conjecture relating the quantum and geometric worlds. Namely:

Conjecture: Let be a long knot with non trivial Vassiliev invariant . Then R(1)(rot(K)-hat(K))=0 if and only if is a torus knot or a satellite with all pieces in the JSJ-decomposition of the complement are Seifert fibered.

The vague dream hiding in the wings is that there is a formula relating vales of to the hyperbolic volume of the knot complement. Even more vaguely, how much geometry of the knot complement does actually see, and how?

R(1) seems genuinely new, deep, and worthy of substantial further study. This is tremendously exciting!

Kea works on the intersection of higher category theory and particle physics, which is niche mathematics combined with niche physics, and as a result has been out of a job for a long time. Marni’s a survivor though (a famous and celebrated survivor, who, together with Sonja Rendell, survived a mountaineering mishap which would have killed the vast majority of us) and she’s been publishing on viXra and continuing to do physics with no funding and often in total abject poverty. It appears to be taking its toll.

I met Marni in New Zealand in 2006 at a Geometrization conference in New Zealand, attended by many celebrated mathematicians and physicists. My impression was of a competent and original physicist. But her work isn’t really valued as far as I can tell. It’s people like Kea who really need and deserve patrons in the old sense of the word, so that they can carry out original and non-mainstream research. If I had a million pounds, I would surely fund Marni’s research. If you’re reading this and you have a million pounds, or if you know somebody with a million pounds, or you run a physics department and want an original outside-the-box professor, I think that to save Kea would be a good thing to do.

The degeneration ratio of a pair of knots , is , the complement of the ratio between the the tunnel number of the connect sum and the sum of the tunnel numbers of the original knots. In other words, this is the amount that the tunnel number drops from the expected amount, divided by the expected amount. Nogueira constructs knot pairs with tunnel numbers three and two, respectively, whose connect sum has tunnel-number three. Thus the degeneration ratio is , which is the highest value known for knots.

As I mentioned in the last post, Trent Schirmer found examples of pairs of two-component links such that the degeneration ratio under connect sums approaches . This is slightly higher than , but it turns out that getting the result for knots instead of links was a highly non-trivial problem. (Schirmer and Nogueira independently discovered roughly the same construction, but Trent wasn’t able to get it to work for knots.) Both of the examples use a structure called a free decomposition that was introduced by Tsuyoshi Kobayashi: Given a knot in a manifold , a free decomposition for is a collection of closed, embedded surface in , transverse to such that the complement in of is a collection of handlebodies. A bridge surface for is one example of a free decomposition, but in general free decompositions can be much more complicated. For example, it is not too difficult to construct a knot that has a free decomposition consisting of surfaces that are incompressible in the complement of the knot – You just have to arrange things so that the compressing disks for the handlebodies in the complement of  must all run along arcs of .

The main difficulty in proving the degeneration ratio for Nogueira’s examples is showing that the original knots have tunnel number three and two, respectively. In general, there are not a lot of good methods for bounding tunnel number from below. (This is essentially the same problem as bounding Heegaard genus from below, which was one of the main obstacles to the rank vs. genus problem.) So, Nogueira’s paper includes a lot of very intricate work characterizing how Heegaard surfaces can intersect these particular free decompositions.

The second paper I mentioned above gives a new bound on tunnel number degeneration. Yo’av Rieck pointed out in the comments to the last post that there are already bounds in certain cases: Morimoto-Schultens have shown [3] that if and are small knots (i.e. there are no closed incompressible surfaces in their complements other than the boundary parallel torus) then the degeneration ratio is zero (the tunnel number of the connect sum is the sum of the original tunnel numbers.) Kobayashi-Rieck generalized this in [4], proving that the same bound holds if both knots are m-small, which means that for any incompressible surface in the complement of the knot, there is a compressing disk for the surface that intersects the knot in a single point (with merdional slope).

As shown by Nogueira’s examples (as well as by the first examples of tunnel number degeneration by Morimoto [5]), this bound can’t possibly hold in the case where one or both knots are not m-small. Thus Trent proves in [2] that if only one of , or is m-small then the tunnel number of the connect sum is greater than or equal to the maximum of the tunnel numbers of and . Translated into the language of degeneration ratio, this means that the degeneration ratio is at most . Trent’s proof again uses free decompositions, by noting that if the tunnel number drops under the connect sum then the resulting Heegaard splitting for the connect sum complement defines a free decomposition for each of the original knots. Trent uses these particular free decompositions to construct Heegaard splittings for the original knots whose genus is less than or equal to the Heegaard splitting of the connect sum complement. He also uses a clever trick to get a similar bound for gluing general three-manifolds along tori, but I’ll let you read his paper for the details of that one.

[3] Morimoto, Kanji; Schultens, Jennifer: Tunnel numbers of small knots do not go down under connected sum. Proc. Amer. Math. Soc. 128 (2000), no. 1, 269-278.

[4] Kobayashi, Tsuyoshi; Rieck, Yo’av: Heegaard genus of the connected sum of m-small knots. Comm. Anal. Geom. 14 (2006), no. 5, 1037-1077.

[5] K. Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math.
Soc. Vol. 123 No. 11 (1995), 3527-3532.

First, you probably noticed that the title of the new blog is an allusion to/rip off of the title of Jeff Week’s (awesome) book The Shape of Space. This book explains the fundamentals of three-dimensional hyperbolic geometry (a la Thurston) in a way that any reasonably bright adult (and many teenagers) can understand. Today, all the attention on “BIG DATA” makes for a great opportunity to introduce the public (including potential future mathematicians) to the geometry that arises naturally in data analysis. (For example high-dimensional data is the perfect motivation for studying higher dimensional spaces!) With the Shape of Data, I hope to explain the geometry of data analysis at a similar level of difficulty as Jeff Weeks’ book.

As I mentioned in my previous post, I think that the field of topology is moving into a stage (experienced by many fields) where it will be enriched by deep connections to applied mathematics. In addition to analysis, there are now applications of topology to DNA knotting and even robotics. I am not suggesting that every topologist should start working on data analysis – It’s not like all number theorists work on encryption or all analysts work on physics. However, both number theory and analysis have benefited – directly and indirectly – from connections to applied mathematics. I hope that as the role played by topology in data analysis grows, it will lead to both greater public interest (and understanding) of the field, and new and interesting problems to work on.

I also think that the field(s) of data analysis can benefit tremendously from a firmer foundation in geometry and topology. We all know that statistics can very easily mislead. As the data gets more complex, the interpretations rely more and more heavily on geometry (even if they do so only implicitly) and misleading statistics can often be caused by a misuse of geometry. (This may sound crazy, but read my upcoming blog posts if you don’t believe me.) The better that we – academics, experts and the general public – understand the way that data is analyzed, the easier it will be to spot misleading statistics, including when “experts” use statistics to lie. I believe that geometry can foster a very intuitive understanding of data analysis and this is what I hope to demonstrate with my new blog.

-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture.

This is big news. I feel it’s the last nail in the coffin of the Hauptvermutung. I’d like to tell you a little bit about the conjecture, and about Manolescu’s strategy, and what it has to do with low dimensional topology.

What is an -manifold?

“That’s a silly question,” someone might say. “It’s a (bla bla bla) space that is locally homeomorphic to ”.

Ah, but homeomorphic via what kind of map? A continuous map? A smooth map? A piecewise linear map? In two dimensions it makes no difference. But in higher dimensions, the concept of an -manifold turns out to be more subtle than the case might lead you to believe.

For Poincaré in 1895, a `three manifold’ starts out as a quotient of by actions of certain groups with a cube as fundamental region. Later in Analysis Situs, he presents a more general notion of a manifold by identifying faces of polyhedra. Poincaré’s manifolds were supposed to be differentiable, but his constructions of them are PL. It seems that Poincaré considered that there was no difference. Obviously all corners can be smoothed and all smooth structures can be approximated by broken linear structures, and (although this notion came later) obviously topological manifolds can all be triangulated. Obvious, but false. The shocking truth is that smooth, PL, and topological categories diverge sharply, even if we restrict ourselves to thinking about individual manifolds and not maps between manifolds.

The following explanation is distilled from Ranicki’s excellent survey On the Hauptvermutung.

A triangulation of a topological space is a simplicial complex together with a homeomorphism from the polyhedron of to . A space is triangulable if it admits a triangulation. In 1908, Steinitz and (independently) Tietze formulated the Main Conjecture, or Hauptvermutung, which states that triangulations of homeomorphic spaces are combinatorially equivalent i.e. become isomorphic after subdivision. In 1961, Milnor shocked the mathematical world by finding a counterexample. But his counterexamples were not manifolds- and as geometric topologists, we’re only supposed to care about manifolds, right? (I really needed sarcastices for that last sentence). So is the Hauptvermutung at least true for manifolds? Closely related, we have the:

Combinatorial Triangulation Conjecture
Every compact topological manifold can be triangulated by a PL manifold.

In dimension two the conjecture holds by 1925 work of Radó, and in dimension three by work of Moise. But (prepare to be shocked again) it’s false in dimensions by work of Kirby and Seibenmann, and in dimension by work of Freedman.

Poor Hauptvermutung! Can’t it at least be just a little bit right?

Let’s try again… Maybe to be homeomorphic to a PL manifold is too much to ask. Recall that the link of a simplex σ is the complex of all simplices at `distance one’ from σ (i.e. separated by one edge from σ). A PL manifold has the property that the link of each simplex of its triangulation is a sphere. Maybe this is too much to ask. After all, there are plenty of not-terribly-pathological spaces which don’t satisfy this property, such as the double suspension of any non-trivial homology sphere (such as the Poincaré sphere), by Cannon and Edwards’s Double Suspension Theorem. So let’s weaken the Combinatorial Triangulation Conjecture:

Triangulation Conjecture
Every compact topological manifold can be triangulated by a locally finite simplicial complex.

Unfortunately even this fails in dimension four, with Freedman’s -manifold providing a counterexample… but maybe dimension four is pathological. Maybe it holds in dimension greater than five.

For all the seeming simplicity of the statement, there’s another surprise in store- even though it’s about manifolds in dimension , it actually reduces to a problem in low dimensional topology of smooth three manifolds. It’s a problem about holomogy cobordism of homology spheres, so let’s briefly review that story.

By the way, the paper I first learnt this story from (and which made a deep impression on me) is:

Ruberman, D., & Saveliev, N. (2005). Casson-type invariants in dimension four. Geometry and topology of manifolds, Fields Inst. Commun, 47, 281-306.

Recall that an integral homology sphere is a closed oriented -manifold with . There are plenty of these, the most famous perhaps being the Poincaré homology sphere. A homology cobordism between homology spheres and is a compact oriented -manifold with boundary such that the inclusions induce isomorphisms for . Homology cobordism is an equivalence relation, and the set of equivalence classes with operation the connect sum forms a group .

Next let’s recall the Rokhlin Invariant. Every integral homology sphere is the boundary of a compact spin -manifold , and the Rokhlin invariant is an eighth of the signature of modulo two. It is invariant under homology cobordism, and so defines a homomorphism .

Alright now! Galewski and Stern, and independently Matsumoto, reduced the Triangulation Conjecture to the following:

Equivalent Statement to the Triangulation Conjecture
There exists a homology -sphere with Rokhlin invariant one which is of order two in .

So the goal becomes to understand torsion in (at least for Rokhlin invariant one homology spheres). It turns out not to be easy, probably because it’s a statement about the smooth category, and so many of our techniques are secretly PL. Until the 1980′s, all that was known about was that is an epimorphism. Then, using equivariant gauge theory, Fintushel and Stern showed that has many elements of infinite order, and Furuta showed that it is infinitely generated. But does it have -torsion? How would you even approach a question like that?

Manolescu’s answer is to construct a Seiberg–Witten Floer Homology with all of the symmetries that the Seiberg-Witten equations have in the situation at hand. Namely, because the bounded four manifold has a spin structure, the Seiberg–Witten equations turn out to have a symmetry group known as . Manolescu has experience with equivariant Seiberg–Witten theories, and he constructs a weapon which (to my untrained eye) looks big enough and strong enough for the task. There’s probably a lot more that this Floer homology can do, and one can now expect vigourous progress in its study.

All of this leaves the Hauptvermutung pretty much as dead as dead can be. Topological manifolds in dimension greater than three just can’t be triangulated in general… And we’re going to have to learn to live with it.

Edit: As pointed out in the comments, by Galewski-Stern dimension 5 counterexamples must be non-orientable (which is quite striking!), but in dimension 6 and above, there are orientable counterexamples as well.

Executive summary for the casual mathematical tourist

Given a lego set whose blocks are triangles (i.e. simplices), there are many shapes (i.e. compact topological manifolds) you could build. But could you build all of them?

In dimension 1 you obviously could (a circle can be cut into line segments), in dimension 2 you obviously could (any compact surface can be sliced up into triangles, and you take those to be the blocks). It turns out that you can in dimension 3 as well (intuitively unsurprising, but difficult to prove), but surprisingly you cannot in dimension 4. But we all know that dimension 4, being the dimension in which we live (space plus time) is a bit crazy- what about in dimension 5 and higher? Galewski and Stern showed that you can triangulate any shape (shape= compact topological manifold) in dimension 5 and up (in just about the weakest sense that a decomposition into triangles has any right to call itself a triangulation) if and only if you can triangulate a specific 5-dimensional shape called the Galewski-Stern manifold. Manolescu’s preprint proves you cannot, showing us again that there are more things in heaven and earth, Horacio, than are dreamt of in your (triangulated) philosophy.

The four grey quadrilaterals on the left indicate four of these bands. Where they come together, the horizontal edge of each band is contained in the union of the horizontal edges of either one or two other bands. The horizontal edges at the top and bottom of the figure would have to be contained in the edges of other bands, not shown.

We will say that a loop (or disjoint union of loops) in a surface is carried by a train track in if the loop is contained in the union of the bands and intersects each band in a collection of vertical arcs (i.e. arcs that are isotopic to for some .)

For example, given a normal loop with respect to a triangulation, part of which is shown in the middle of the figure, we can construct a train track by placing three bands in each triangle, one for each family of normal arcs. We can place the horizontal edges of each band in the edges of the triangulation, as on the right, so that the normal loop is carried by the train track. Notice that the way that the horizontal edges of the band are matched up is determined by the loop. In this case, the fact that there are edges going from the upper left edge of these two triangles to the lower right edge means that the upper left band should meet the lower right band along its lower horizontal edge.

A normal loop that has arcs going from the lower left to upper right edges would not be carried by this train track. This illustrates the first big difference between normal loops and train tracks: Usually, a given train track will not carry every isotopy class of essential loops in . We could build a different train track that would carry this second type of loop, but then it wouldn’t carry the loop that’s shown. However, this turns out to be a small price to pay for the second difference between normal loops and train tracks: Generally, a train track will not carry any isotopy trivial loops, and moreover will carry at most one representative of any isotopy class of loops. As a third difference, train tracks are much more flexible because we can put together the bands in ways that are not necessarily induced by a triangulation.

What train tracks and normal loops have in common is the vector space structure: Every loop carried by a train track is completely determined by a vector of integers that count how many times the loop intersects each band in the train track. And as with normal loops, adding vectors corresponds to the same, very simple geometric operation called a Haken sum. So both a triangulation and a train track define a map from a vector space into the set of loops in a surface . The map from the set of normal loops is onto, but is many-to-one and its image contains the trivial loop. The map from the loops carried by the train track is not onto, but it is one-to-one (assuming some simple conditions on the train track) and only maps vectors to essential loops. Moreover, if we consider a large enough set of different train tracks, the images of all these maps will be onto the set of essential loops. So we can think of the train tracks as defining patches of loops, reminiscent of the patches in Reimannian geometry that allow one to piece together a manifold from Euclidean balls. In this case, the patches define a local vector-space structure on the set of loops that can be used, for example, to define the space of projective measured laminations on . But that will have to wait for a future post.