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The other day at lunch, one of my colleagues — let’s call her “Wendy Hilton” to preserve her anonymity (OK, this is pretty bad, but perhaps not quite as bad as Clive James’s use of “Romaine Rand” as a pseudonym for “Germaine Greer” in Unreliable Memoirs . . .) — expressed some skepticism about a somewhat unusual assertion that I make at the start of my scl monograph. Since it is my monograph, I feel free to quote the offending paragraphs:

It is unfortunate in some ways that the standard way to refer to the plane emphasizes its product structure. This product structure is topologically unnatural, since it is deﬁned in a way which breaks the natural topological symmetries of the object in question. This fact is thrown more sharply into focus when one discusses more rigid topologies.

At this point I give an example, namely that of the Zariski topology, pointing out that the product topology of two copies of the affine line with the Zariski topology is not the same as the Zariski topology on the affine plane. All well and good. I then go on to claim that part of the bias is biological in origin, citing the following example as evidence:

Example 1.2 (Primary visual cortex). The primary visual cortex of mammals (including humans), located at the posterior pole of the occipital cortex, contains neurons hardwired to ﬁre when exposed to certain spatial and temporal patterns. Certain speciﬁc neurons are sensitive to stimulus along speciﬁc orientations, but in primates, more cortical machinery is devoted to representing vertical and horizontal than oblique orientations (see for example [58] for a discussion of this eﬀect).

(Note: [58] is a reference to the paper “The distribution of oriented contours in the real world” by David Coppola, Harriett Purves, Allison McCoy, and Dale Purves, Proc. Natl. Acad. Sci. USA 95 (1998), no. 7, 4002–4006)

I think Wendy took this to be some kind of poetic license or conceit, and perhaps even felt that it was a bit out of place in a serious research monograph. On balance, I think I agree that it comes across as somewhat jarring and unexpected to the reader, and the tone and focus is somewhat inconsistent with that of the rest of the book. But I also think that in certain subjects in mathematics — and I would put low-dimensional geometry/topology in this category — we are often not aware of the extent to which our patterns of reasoning and imagination are shaped, limited, or (mis)directed by our psychological — and especially psychophysical — natures.

The particular question of how the mind conceives of, imagines, or perceives any mathematical object is complicated and multidimensional, and colored by historical, social, and psychological (not to mention mathematical) forces. It is generally a vain endeavor to find precise physical correlates of complicated mental objects, but in the case of the plane (or at least one cognitive surrogate, the subjective visual field) there is a natural candidate for such a correlate. Cells on the rear of the occipital lobe are arranged in a “map” in the region of the occipital lobe known as the “primary visual cortex”, or V1. There is a precise geometric relationship between the location of neurons in V1 and the points in the subjective visual field they correspond to. Further visual processing is done by other areas V2, V3, V4, V5 of the visual cortex. Information is fed forward from Vi to Vj with $j>i$, but also backward from Vj to Vi regions, so that visual information is processed at several levels of abstraction simultaneously, and the results of this processing compared and refined in a complicated synthesis (this tends to make me think of the parallel terraced scan model of analogical reasoning put forward by Douglas Hofstadter and Melanie Mitchell; see Fluid concepts and creative analogies, Chapter 5).

The initial processing done by the V1 area is quite low-level; individual neurons are sensitive to certain kind of stimuli, e.g. color, spatial periodicity (on various scales),  motion, orientation, etc. As remarked earlier, more neurons are devoted to detecting horizontally or vertically aligned stimuli; in other words, our brains literally devote more hardware to perceiving or imagining vertical and horizontal lines than to lines with an oblique orientation. This is not to say that at some higher, more integrated level, our perception is not sensitive to other symmetries that our hardware does not respect, just as a random walk on a square lattice in the plane converges (after a parabolic rescaling) to Brownian motion (which is not just rotationally but conformally invariant). However the fact is that humans perform statistically better on cognitive tasks that involve the perception of figures that are aligned along the horizontal and vertical axes, than on similar tasks that differ only by a rotation of the figures.

It is perhaps interesting therefore that the earliest (?) mathematical conception of the plane, due to the Greeks, did not give a privileged place to the horizontal or vertical directions, but treats all orientations on an equal footing. In other words, in Greek (Euclidean) geometry, the definitions respect the underlying symmetries of the objects. Of course, from our modern perspective we would not say that the Greeks gave a definition of the plane at all, or at best, that the definition is woefully inadequate. According to one well-known translation, the plane is introduced as a special kind of surface as follows:

A surface is that which has length and breadth.

When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.

This definition of a surface looks as though it is introducing coordinates, but in fact one might just as well interpret it as defining a surface in terms of its dimension; having defined a surface (presumably thought of as being contained in some ambient undefined three-dimensional space) one defines a plane to be a certain kind of surface, namely one that is convex. Horizontal and vertical axes are never introduced. Perpendicularity is singled out as important, but the perpendicularity of two lines is a relative notion, whereas horizontality and verticality are absolute. In the end, Euclidean geometry is defined implicitly by its properties, most importantly isotropy (i.e. all right angles are equal to one another) and the parallel postulate, which singles it out from among several alternatives (elliptic geometry, hyperbolic geometry). In my opinion, Euclidean geometry is imprecise but natural (in the sense of category theory), because objects are defined in terms of the natural transformations they admit, and in a way that respects their underlying symmetries.

In the 15th century, the Italian artists of the Renaissance developed the precise geometric method of perspective painting (although the technique of representing more distant objects by smaller figures is extremely ancient). Its invention is typically credited to the architect and engineer Filippo Brunelleschi; one may speculate that the demands of architecture (i.e. the representation of precise 3 dimensional geometric objects in 2 dimensional diagrams) was one of the stimuli that led to this invention (perhaps this suggestion is anachronistic?). Mathematically, this gives rise to the geometry of the projective plane, i.e. the space of lines through the origin (the “eye” of the viewer of a scene). In principle, one could develop projective geometry without introducing “special” directions or families of lines. However, in one, two, or three point perspective, families of lines parallel to one or several “special” coordinate axes (along which significant objects in the painting are aligned) appear to converge to one of the vanishing points of the painting. In his treatise “De pictura” (on painting), Leon Battista Alberti (a friend of Brunelleschi) explicitly described the geometry of vision in terms of projections on to a (visual) plane. Amusingly (in the context of this blog post), he explicitly distinguishes between the mathematical and the visual plane:

In all this discussion, I beg you to consider me not as a mathematician but as a painter writing of these things.

Mathematicians measure with their minds alone the forms of things separated from all matter. Since we wish the object to be seen, we will use a more sensate wisdom.

I beg to differ: similar parts of the brain are used for imagining a triangle and for looking at a painting. Alberti’s claim sounds a bit too much like Gould’s “non-overlapping magisteria”, and in a way it is disheartening that it was made at a place and point in history at which mathematics and the visual arts were perhaps at their closest.

In the 17th century René Descartes introduced his coordinate system and thereby invented “analytic geometry”. To us it might not seem like such a big leap to go from a checkerboard floor in a perspective painting (or a grid of squares to break up the visual field) to the introduction of numerical coordinates to specify a geometrical figure, but Descartes’s ideas for the first time allowed mathematicians to prove theorems in geometry by algebraic methods. Analytic geometry is contrasted with “synthetic geometry”, in which theorems are deduced logically from primitive axioms and rules of inference. In some abstract sense, this is not a clear distinction, since algebra and analysis also rests on primitive axioms, and rules of deduction. In my opinion, this terminology reflects a psychological distinction between “analytic methods” in which one computes blindly and then thinks about what the results of the calculation mean afterwards, and “synthetic methods” in which one has a mental model of the objects one is manipulating, and directly intuits the “meaning” of the operations one performs. Philosophically speaking, the first is formal, the second is platonic. Biologically speaking, the first does not make use of the primary visual cortex, the second does.

As significant as Descartes ideas were, mathematicians were slow to take real advantage of them. Complex numbers were invented by Cardano in the mid 16th century, but the idea of representing complex numbers geometrically, by taking the real and imaginary parts as Cartesian coordinates, had to wait until Argand in the early 19th.

Incidentally, I have heard it said that the Greeks did not introduce coordinates because they drew their figures on the ground and looked at them from all sides, whereas Descartes and his contemporaries drew figures in books. Whether this has any truth to it or not, I do sometimes find it useful to rotate a mathematical figure I am looking at, in order to stimulate my imagination.

After Poincaré’s invention of topology in the late 19th century, there was a new kind of model of the plane to be (re)imagined, namely the plane as a topological space. One of the most interesting characterizations was obtained by the brilliantly original and idiosyncratic R. L. Moore in his paper, “On the foundations of plane analysis situs”. Let me first remark that the line can be characterized topologically in terms of its natural order structure; one might argue that this characterization more properly determines the oriented line, and this is a fair comment, but at least the object has been determined up to a finite ambiguity. Let me second of all remark that the characterization of the line in terms of order structures is useful; a (countable) group $G$ is abstractly isomorphic to a group of (orientation-preserving) homeomorphisms of the line if and only if $G$ admits an (abstract) left-invariant order.

Given points and the line, Moore proceeds to list a collection of axioms which serve to characterize the plane amongst topological spaces. The axioms are expressed in terms of separation properties of primitive undefined terms called points and regions (which correspond more or less to ordinary points and open sets homeomorphic to the interiors of closed disks respectively) and non-primitive objects called “simple closed curves” which are (eventually) defined in terms of simpler objects. Moore’s axioms are “natural” in the sense that they do not introduce new, unnecessary, unnatural structure (such as coordinates, a metric, special families of “straight” lines, etc.). The basic principle on which Moore’s axioms rest is that of separation — which continua separate which points from which others? If there is a psychophysical correlate of this mathematical intuition, perhaps it might be the proliferation of certain neurons in the primary visual cortex which are edge detectors — they are sensitive, not to absolute intensity, but to a spatial discontinuity in the intensity (associated with the “edge” of an object). The visual world is full of objects, and our eyes evolved to detect them, and to distinguish them from their surroundings (to distinguish figure from ground as it were). If I have an objection to Cartesian coordinates on biological grounds (I don’t, but for the sake of argument let’s suppose I do) then perhaps Moore should also be disqualified for similar reasons. Or rather, perhaps it is worth being explicitly aware, when we make use of a particular mathematical model or intellectual apparatus, of which aspects of it are necessary or useful because of their (abstract) applications to mathematics, and which are necessary or useful because we are built in such a way as to need or to be able to use them.

Last Friday, Henry Wilton gave a talk at Caltech about his recent joint work with Sang-hyun Kim on polygonal words in free groups. Their work is motivated by the following well-known question of Gromov:

Question(Gromov): Let $G$ be a one-ended word-hyperbolic group. Does $G$ contain a subgroup isomorphic to the fundamental group of a closed hyperbolic surface?

Let me briefly say what “one-ended” and “word-hyperbolic” mean.

A group is said to be word-hyperbolic if it acts properly and cocompactly by isometries on a proper $\delta$-hyperbolic path metric space — i.e. a path metric space in which there is a constant $\delta$ so that geodesic triangles in the metric space have the property that each side of the triangle is contained in the $\delta$-neighborhood of the union of the other two sides (colloquially, triangles are thin). This condition distills the essence of negative curvature in the large, and was shown by Gromov to be equivalent to several other conditions (eg. that the group satisfies a linear isoperimetric inequality; that every ultralimit of the group is an $\mathbb{R}$-tree). Free groups are hyperbolic; fundamental groups of closed manifolds with negative sectional curvature (eg surfaces with negative Euler characteristic) are word-hyperbolic; “random” groups are hyperbolic — and so on. In fact, it is an open question whether a group $G$ that admits a finite $K(G,1)$ is word hyperbolic if and only if it does not contain a copy of a Baumslag-Solitar group $BS(m,n):=\langle x,y \; | \; x^{-1}y^{m}x = y^n \rangle$ for $m,n \ne 0$ (note that the group $\mathbb{Z}\oplus \mathbb{Z}$ is the special case $m=n=1$); in any case, this is a very good heuristic for identifying the word-hyperbolic groups one typically meets in examples.

If $G$ is a finitely generated group, the ends of $G$ really means the ends (as defined by Freudenthal) of the Cayley graph of $G$ with respect to some finite generating set. Given a proper topological space $X$, the set of compact subsets of $X$ gives rise to an inverse system of inclusions, where $X-K'$ includes into $X-K$ whenever $K$ is a subset of $K'$. This inverse system defines an inverse system of maps of discrete spaces $\pi_0(X-K') \to \pi_0(X-K)$, and the inverse limit of this system is a compact, totally disconnected space $\mathcal{E}(X)$, called the space of ends of $X$. A proper topological space is canonically compactified by its set of ends; in fact, the compactification $X \cup \mathcal{E}(X)$ is the “biggest” compactification of $X$ by a totally disconnected space, in the sense that for any other compactification $X \subset Y$ where $Y-X$ is zero dimensional, there is a continuous map $X \cup \mathcal{E}(X) \to Y$ which is the identity on $X$.

For a word-hyperbolic group $G$, the Cayley graph can be compactified by adding the ideal boundary $\partial_\infty G$, but this is typically not totally disconnected. In this case, the ends of $G$ can be recovered as the components of $\partial_\infty G$.

A group $G$ acts on its own ends $\mathcal{E}(G)$. An elementary argument shows that the cardinality of $\mathcal{E}(G)$ is one of $0,1,2,\infty$ (if a compact set $V$ disconnects $e_1,e_2,e_3$ then infinitely many translates of $V$ converging to $e_1$ separate $e_3$ from infinitely many other ends accumulating on $e_1$). A group has no ends if and only if it is finite. Stallings famously showed that a (finitely generated) group has at least $2$ ends if and only if it admits a nontrivial description as an HNN extension or amalgamated free product over a finite group. One version of the argument proceeds more or less as follows, at least when $G$ is finitely presented. Let $M$ be an $n$-dimensional Riemannian manifold with fundamental group $G$, and let $\tilde{M}$ denote the universal cover. We can identify the ends of $G$ with the ends of $\tilde{M}$. Let $V$ be a least ($n-1$-dimensional) area hypersurface in $\tilde{M}$ amongst all hypersurfaces that separate some end from some other (here the hypothesis that $G$ has at least two ends is used). Then every translate of $V$ by an element of $G$ is either equal to $V$ or disjoint from it, or else one could use the Meeks-Yau “roundoff trick” to find a new $V'$ with strictly lower area than $V$. The translates of $V$ decompose $\tilde{M}$ into pieces, and one can build a tree $T$ whose vertices correspond to to components of $\tilde{M} - G\cdot V$, and whose edges correspond to the translates $G\cdot V$. The group $G$ acts on this tree, with finite edge stabilizers (by the compactness of $V$), exhibiting $G$ either as an HNN extension or an amalgamated product over the edge stabilizers. Note that the special case $|\mathcal{E}(G)|=2$ occurs if and only if $G$ has a finite index subgroup which is isomorphic to $\mathbb{Z}$.

Free groups and virtually free groups do not contain closed surface subgroups; Gromov’s question more or less asks whether these are the only examples of word-hyperbolic groups with this property.

Kim and Wilton study Gromov’s question in a very, very concrete case, namely that case that $G$ is the double of a free group $F$ along a word $w$; i.e. $G = F *_{\langle w \rangle } F$ (hereafter denoted $D(w)$). Such groups are known to be one-ended if and only if $w$ is not contained in a proper free factor of $F$ (it is clear that this condition is necessary), and to be hyperbolic if and only if $w$ is not a proper power, by a result of Bestvina-Feighn. To see that this condition is necessary, observe that the double $\mathbb{Z} *_{p\mathbb{Z}} \mathbb{Z}$ is isomorphic to the fundamental group of a Seifert fiber space, with base space a disk with two orbifold points of order $p$; such a group contains a $\mathbb{Z}\oplus \mathbb{Z}$. One might think that such groups are too simple to give an insight into Gromov’s question. However, these groups (or perhaps the slightly larger class of graphs of free groups with cyclic edge groups) are a critical case for at least two reasons:

1. The “smaller” a group is, the less room there is inside it for a surface group; thus the “simplest” groups should have the best chance of being a counterexample to Gromov’s question.
2. If $G$ is word-hyperbolic and one-ended, one can try to find a surface subgroup by first looking for a graph of free groups $H$ in $G$, and then looking for a surface group in $H$. Since a closed surface group is itself a graph of free groups, one cannot “miss” any surface groups this way.

Not too long ago, I found an interesting construction of surface groups in certain graphs of free groups with cyclic edge groups. In fact, I showed that every nontrivial element of $H_2(G;\mathbb{Q})$ in such a group is virtually represented by a sum of surface subgroups. Such surface subgroups are obtained by finding maps of surface groups into $G$ which minimize the Gromov norm in their (projective) homology class. I think it is useful to extend Gromov’s question by making the following

Conjecture: Let $G$ be a word-hyperbolic group, and let $\alpha \in H_2(G;\mathbb{Q})$ be nonzero. Then some multiple of $\alpha$ is represented by a norm-minimizing surface (which is necessarily $\pi_1$-injective).

Note that this conjecture does not generalize to wider classes of groups. There are even examples of $\text{CAT}(0)$ groups $G$ with nonzero homology classes $\alpha \in H_2(G;\mathbb{Q})$ with positive, rational Gromov norm, for which there are no $\pi_1$-injective surfaces representing a multiple of $\alpha$ at all.

It is time to define polygonal words in free groups.

Definition: Let $F$ be free. Let $X$ be a wedge of circles whose edges are free generators for $F$. A cyclically reduced word $w$ in these generators is polygonal if there exists a van-Kampen graph $\Gamma$ on a surface $S$ such that:

1. every complementary region is a disk whose boundary is a nontrivial (possibly negative) power of $w$;
2. the (labelled) graph $\Gamma$ immerses in $X$ in a label preserving way;
3. the Euler characteristic of $S$ is strictly less than the number of disks.

The last condition rules out trivial examples; for example, the double of a single disk whose boundary is labeled by $w^n$. Notice that it is very important to allow both positive and negative powers of $w$ as boundaries of complementary regions. In fact, if $w$ is not in the commutator subgroup, then the sum of the powers over all complementary regions is necessarily zero (and if $w$ is in the commutator subgroup, then $D(w)$ has nontrivial $H_2$, so one already knows that there is a surface subgroup).

Condition 2. means that at each vertex of $\Gamma$, there is at most one oriented label corresponding to each generator of $F$ or its inverse. This is really the crucial geometric property. If $\Gamma,S$ is a van-Kampen graph as above, then a theorem of Marshall Hall implies that there is a finite cover of $X$ into which $\Gamma$ embeds (in fact, this observation underlies Stallings’s work on foldings of graphs). If we build a $2$-complex $Y$ with $\pi_1(Y)=D(w)$ by attaching two ends of a cylinder to suitable loops in two copies of $X$, then a tubular neighborhood of $\Gamma$ in $S$ (i.e. what is sometimes called a “fatgraph” ) embeds in a finite cover $\tilde{Y}$ of $Y$, and its double — a surface of strictly negative Euler characteristic — embeds as a closed surface in $\tilde{Y}$, and is therefore $\pi_1$-injective. Hence if $w$ is polygonal, $D(w)$ contains a surface subgroup.

Not every word is polygonal. Kim-Wilton discuss some interesting examples in their paper, including:

1. suppose $w$ is a cyclically reduced product of proper powers of the generators or their inverses (e.g a word like $a^3b^7a^{-2}c^{13}$ but not a word like $a^3bc^{-1}$); then $w$ is polygonal;
2. a word of the form $\prod_i a^{p_{2i-1}}(a^{p_{2i}})^b$ is polygonal if $|p_i|>1$ for each $i$;
3. the word $abab^2ab^3$ is not polygonal.

To see 3, suppose there were a van-Kampen diagram with more disks than Euler characteristic. Then there must be some vertex of valence at least $3$. Since $w$ is positive, the complementary regions must have boundaries which alternate between positive and negative powers of $w$, so the degree of the vertex must be even. On the other hand, since $\Gamma$ must immerse in a wedge of two circles, the degree of every vertex must be at most $4$, so there is consequently some vertex of degree exactly $4$. Since each $a$ is isolated, at least $2$ edges must be labelled $b$; hence exactly two. Hence exactly two edges are labelled $a$. But one of these must be incoming and one outgoing, and therefore these are adjacent, contrary to the fact that $w$ does not contain a $a^{\pm 2}$.

1 above is quite striking to me. When $w$ is in the commutator subgroup, one can consider van-Kampen diagrams as above without the injectivity property, but with the property that every power of $w$ on the boundary of a disk is positive; call such a van-Kampen diagram monotone. It turns out that monotone van-Kampen diagrams always exist when $w \in [F,F]$, and in fact that norm-minimizing surfaces representing powers of the generator of $H_2(D(w))$ are associated to certain monotone diagrams. The construction of such surfaces is an important step in the argument that stable commutator length (a kind of relative Gromov norm) is rational in free groups. In my paper scl, sails and surgery I showed that monomorphisms of free groups that send every generator to a power of that generator induce isometries of the $\text{scl}$ norm; in other words, there is a natural correspondence between certain equivalence classes of monotone surfaces for an arbitrary word in $[F,F]$ and for a word of the kind that Kim-Wilton show is polygonal (Note: Henry Wilton tells me that Brady, Forester and Martinez-Pedroza have independently shown that $D(w)$ contains a surface group for such $w$, but I have not seen their preprint (though I would be very grateful to get a copy!)).

In any case, if not every word is polygonal, all is not lost. To show that $D(w)$ contains a surface subgroup is suffices to show that $D(w')$ contains a surface subgroup, where $w$ and $w'$ differ by an automorphism of $F$. Kim-Wilton conjecture that one can always find an automorphism $\phi$ so that $\phi(w)$ is polygonal. In fact, they make the following:

Conjecture (Kim-Wilton; tiling conjecture): A word $w$ not contained in a proper free factor of shortest length (in a given generating set) in its orbit under $\text{Aut}(F)$ is polygonal.

If true, this would give a positive answer to Gromov’s question for groups of the form $D(w)$.

I was at UC Riverside this past weekend, attending the regional meeting, and giving a talk in a special session on knot theory in memory of the late Xiao-Song Lin. After lunch, I joined in a conversation between Rob Kirby and Mike Freedman on the recent flurry of activity this summer, in which Selman Akbulut showed (and his work was further extended by Bob Gompf ) that certain infinite families of Cappell-Shaneson manifolds — smooth $4$-manifolds known since Freedman’s work to be homeomorphic to $S^4$ — are in fact diffeomorphic to the standard smooth $S^4$ (actually, Cappell-Shaneson’s manifolds have the additional feature that they admit a free $\mathbb{Z}/2\mathbb{Z}$ action, giving rise to fake $\mathbb{RP}^4$‘s, which was actually their original interest). (Note: an earlier version of this post falsely implied that Gompf’s work was done independently of Akbulut’s, whereas in fact it came later, as Gompf readily acknowledges).

Apparently these constructions had somewhat altered the experts’s (i.e. Freedman and Kirby) feelings about whether the smooth $4$-dimensional Poincaré conjecture is likely to be true. The Cappell-Shaneson manifolds are constructed by doing surgery on certain torus bundles over a circle — those with monodromy chosen so that the resulting torus bundles have the homology of a $S^1 \times S^3$. A suitable surgery, killing the $S^1$ factor makes the manifold into homology $S^4$‘s, and also kills the subgroup of the fundamental group normally generated by the $S^1$ factor. On the other hand, everything else in the fundamental group “comes from” the torus, whose fundamental group is abelian, and therefore the resulting manifold is simply-connected. Since it is a homology $4$-sphere, is it therefore a homotopy $4$-sphere, and consequently (by Freedman), a topological $4$-sphere.

Gompf shows these $4$-spheres are standard by showing that a certain move — which simplifies the monodromy of the $T^3$ fiber — can be realized by a diffeomorphism. The move is an example of what is known to $4$-manifold topologists as a “log transform” (and to $3$-manifold topologists as “Dehn surgery times $S^1$”). A log transform takes as input a smooth embedded torus. A tubular neighborhood of this torus is a product $T^2 \times D^2$ whose boundary is a $3$-torus $T^3$. This tubular neighborhood is removed, and reglued by an automorphism of the $T^3$ factor. Usually a log transform will change the topology of the manifold, or at least the smooth structure. But in this case, the surgered torus is contained in a $T^3$ fiber, and the log transform can be shown to be isotopic to the identity, by using the monodromy of the fibration (technically, the monodromy of the fibration produces a once-punctured torus in the $T^3$ bundle with boundary on the curve along which the log transform “twists”, but after doing surgery to produce the homology $S^4$‘s, this once-punctured torus is “capped” to become a smooth disk).

The point of this blog post is to show how to construct many, many other smooth $4$-manifolds which are topological $4$-spheres, and for which Gompf’s method of showing they are standard does not work. Are these manifolds counterexamples to the smooth $4$-dimensional Poincaré conjecture? I am really not the person to ask.

The construction takes as input a fibered knot — i.e. a knot $K$ in the $3$-sphere $S^3$ whose complement fibers over a circle. In other words, there is a fibration $S \to S^3 - K \to S^1$, where $S$ is a (minimal genus) Seifert surface for the knot $K$. The fibration of spaces gives rise to a short exact sequence of fundamental groups (in general, one gets a long exact sequence of homotopy groups, but the spaces $S, S^3-K,S^1$ are all $K(\pi,1)$‘s — i.e. their homotopy groups in dimension other than $1$ all vanish). Since $S$ has boundary, the fundamental group of $S$ is free and finitely generated of rank $2g$, where $g$ is the genus. The fundamental group of $S^1$ is $\mathbb{Z}$. So one exhibits $\pi_1(S^3 - K)$ as an HNN extension of a free group, where the meridian $m$ acts by conjugation on the free group $\pi_1(S)$ by some automorphism $\phi:\pi_1(S) \to \pi_1(S)$.

Since $K$ is a knot in $S^3$, the homology of $S^3-K$ is equal to $\mathbb{Z}$ in dimension $1$. Moreover, since putting $K$ back in recovers $S^3$, it follows that the fundamental group $\pi_1(S^3-K)$ is normally generated by the meridian (which also generates the $\mathbb{Z}$ in $H_1$). For the moment everything is $3$-dimensional, but there is a trick to promote this to $4$ dimensions. In place of the surface $S$, consider the $3$-manifold $M_{2g} = \#_{i=1}^{2g} S^2 \times S^1$. In other words, $M_{2g}$ is obtained by doubling a handlebody of genus $2g$. The fundamental group $\pi_1(M_{2g})$ is free of rank $2g$. Now one builds a $M$ bundle over $S^1$ with monodromy $\phi$; call this $4$-manifold $W_\phi$. The existence of such a manifold depends on being able to realize any automorphism of a free group by a homeomorphism of a doubled handlebody; one way to see this is to observe that $\text{Aut}(F)$ is generated by Nielsen moves — interchanging generators, replacing generators by their inverses, and replacing generators $x,y$ by $xy, y$. These moves are all realizable by homeomorphisms of doubled handlebodies, the last by a “handle slide”.

Now, observe that $\pi_1(W_\phi) = \pi_1(S^3 - K)$, and is normally generated by a loop $\gamma \in W_\phi$ representing the circle direction. Moreover, $H_1(W_\phi) = H_1(S^3-K) = \mathbb{Z}$. To compute $H_2$, observe that $H_2(M) = (H_1(M))^*$ by Poincaré duality. If the action of $\phi$ on $H_1(M)$ (a free abelian group of rank $2g$) is represented by a matrix $A$, then the action on $H_2(M)$ is represented by the transpose $A^T$. The fact that $H_1(W_\phi)=\mathbb{Z}$ is equivalent to the fact that the first homology of $M$ dies in the bundle; i.e. $\det(A - \text{Id})=\pm 1$; hence $\det(A^T - \text{Id}) = \pm 1$, and for the same reason, the second homology of $M$ dies in the bundle, and $H_2(W_\phi)=0$. By ($4$-dimensional) Poincaré duality, $H_3(W_\phi) = \mathbb{Z}$, and we see that $W_\phi$ is a homology $S^1 \times S^3$.

A tubular neighborhood of the loop $\gamma$ is a product $S^1 \times D^3$, since $W_\phi$ is orientable. The boundary of this is $S^1 \times S^2$. So we drill out $\gamma$ and glue in a product $D^2 \times S^2$ to produce $W_\phi'$. Drilling out $\gamma$ does not affect the fundamental group, by Seifert van-Kampen, and the fact that the inclusion $S^1 \times S^2 \to S^1 \times D^3$ is an isomorphism on $\pi_1$. On the other hand, filling in a $D^2 \times S^2$ has the effect of killing the meridian $m$, and therefore (by the discussion above), killing $\pi_1$ completely; i.e. $W_\phi'$ is simply-connected. Hence $H_1(W_\phi') = H_3(W_\phi')=0$. Drilling out a circle and gluing back a sphere increases Euler characteristic by $2$; since the rank of $H_1$ and $H_3$ have both gone down by $1$, it follows that the rank of $H_2(W_\phi')$ is still $0$, and since the fundamental group is trivial, $H_2(W_\phi')=0$. So $W_\phi'$ is a smooth, simply-connected homology sphere, which is to say, a smooth $4$-manifold which is topologically $S^4$.

Back in June, Freedman-Gompf-Morrison-Walker described a way to use Rasmussen’s $s$-invariant to detect exotic $S^4$‘s, and proposed trying this invariant out on the Cappell-Shaneson examples (see Scott Morrison’s post about that here). Is it feasible to compute the invariants on these new examples?

(Corrected Update 11/10:) Some ways of doing this construction give standard $S^4$‘s, some give $S^4$‘s that are not obviously standard. And there are other variations on this construction arising from “non-geometric” automorphisms of free groups that are also not obviously standard. These examples are also not obviously the same as other known potential counterexamples to the smooth Poincare conjecture. So the conclusion seems to be that they deserve further study. (Added 11/15:) This paper by Aitchison-Silver discusses a closely related construction.

The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian $2$-manifold $\Sigma$, one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if $\Sigma$ is a covering space of some other Riemannian surface $S$, then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface $S$ (assuming that the metric on $S$ is sufficiently generic; otherwise, it recovers $S$ modulo its group of isometries). Morally what one is doing is mapping $\Sigma$ into the space $\mathcal{M}$ of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in $\mathcal{M}$ is a pair $(X,p)$ where $X$ is a locally compact metric space, and $p \in X$ is a point. A sequence $X_i,p_i$ converges to $X,p$ if there are metric balls $B_i$ around $p_i$ of diameter going to infinity, metric balls $D_i$ around $p$ also of diameter going to infinity, and isometric inclusions of $B_i,D_i$ into metric spaces $Z_i$ in such a way that the Hausdorff distance between the images of $B_i$ and $D_i$ in $Z_i$ goes to zero as $i \to \infty$. Any locally compact metric space $Y$ has a tautological map to $\mathcal{M}$, where each point $y \in Y$ is sent to the point $(Y,y) \in \mathcal{M}$. Gromov showed (see section 6 of this paper) that the space $\mathcal{M}$ itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If $\Sigma$ has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of $\Sigma$ in $\mathcal{M}$ is precompact, and its closure is a compact metric space $\mathcal{L}$. The path components of $\mathcal{L}$ are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of $\Sigma$. If you were wandering around on such a component $\Sigma'$, and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on $\Sigma$. Topologically, $\mathcal{L}$ is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form $U \times X$ where $U$ is an open two-dimensional disk, where $X$ is totally disconnected, and where the transition between charts preserves the decomposition into pieces $U \times \text{point}$, and is smooth (in fact, preserves the Riemann surface structure) on the $U \times \text{point}$ slices, in the overlaps. The unions of “surface” slices — i.e. the path components of $\mathcal{L}$ — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region $K$ in $\Sigma$ and every $\epsilon>0$ there is a $T$ so that every ball in $\Sigma$ of radius $T$ contains a subset $K'$ which is $\epsilon$-close to $K$ in the Gromov-Hausdorff metric. In other words, every “local feature” of $\Sigma$ that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf $\Sigma'$ of $\mathcal{L}$, and therefore $\Sigma$ is in the closure of each $\Sigma'$. Since $\mathcal{L}$ is (in) the closure of $\Sigma$, this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in $\mathcal{L}$. Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a $4$-times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane $\mathbb{C}^*$. Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let $T_1$ be the planar “Greek cross” as in the following figure:

Inductively, if we have defined $T_n$, define $T_{n+1}$ by attaching four copies of $T_n$ to the extremities of $T_1$. The first few examples $T_1,\cdots,T_4$ are illustrated in the following figure:

The limit $T_\infty$ is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points $p_i$ in $T_\infty$ has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination $\mathcal{L}$ this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in $3$-dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize $3$-manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.