Postcard from a Geometric Topologist

Text: Take a 3d real vector space $M$ with a Lorentz form (a symmetric bilinear form with signture +--). Given a 2d real vector space V with a volume form (an alternating bilinear form), we can find a double covering of SO(M) by SL(V). We can make this fact constructive by building V as a double cover of the light cone of M.

Let's work backward. The volume form D on V induces a Lorentz form (vv', ww') = [D(v, w) D(v', w') + D(v, w') D(v', w)] / 4 on Sym^2 V. A basis e, f for V with D(e, f) = 1 gives an orthonormal basis e^2 + f^2 (positive), e^2 - f^2 (negative), 2ef (negative) for Sym^2 V. Since the action of SL(V) on V preserves the volume form, the action of SL(V) on Sym^2 V preserves the Lorentz form. This gives a double covering of SO(Sym^2 V) by SL(V).

The squaring map v maps to v^2 sends V to the forward light cone of Sym^2 V. In fact, it's a double covering of the forward light cone, branched at 0. The line D(v, _) = c$ in V maps to the curve (v^2, _) = c^2 / 2 in the light cone.

As a topological branched cover, we can reconstruct V from the space of paths on the forward light cone. What about the vector space structure?

Given v^2, w^2, we can find (v + w)^2 and (v - w)^2 at the points on the forward light cone where the curves (v^2, _) = (v^2, w^2) and (w^2, _) = (v^2, w^2) intersect. Given a path (v_t)^2, we can distinguish (v_0 + v_t)^2 and (v_0 - v_t)^2 by observing that (v_0 + v_t)^2 starts at 4(v_0)^2, while (v_0 - v_t)^2 starts at 0. This should let us reconstruct V, as a vector space, from the space of paths on the forward light cone.

Image Description: 

The vector space V with the basis e, f and a corresponding coordinate grid.

The light cone in Sym^2 V, with e^2, f^2, and the squared coordinate grid shown on the forward half. The grid lines through the origin of V map to rays in the forward light cone. The other grid lines map to concentric parabolas around these rays.

An arrow from V to the forward light cone, labeled "squaring".

Another view of the forward light cone, with (e + f)^2 and (e - f)^2 shown in addition to everything above. The images of the e-parallel grid line through f and the f-parallel grid line through e form a bigon, with (e + f)^2 and (e - f)^2 at its corners.

Image description: 

A link with black, red, blue, and green components.

Text: 

This picture illustrates why the figure-eight knot is rational homology slice. (A more complicated picture was given by Akbulut-Larson.)

The black, red, and green curves form a Kirby diagram for a rational homology 4-ball X with $\partial X = S^3$ and $H_1(X) = Z/2$.

If we ignore the red and blue 2-handles, we see that the green curve K is unknotted and unlinked from the dotted black curve, so K bounds a smoothly embedded disk in X. Some handleslides of K over the other curves take K to a local figure-eight knot (exercise!).

I am interested in generalizing this picture to find knots that are slice in rational homology balls with H_1 of odd order, but not in the 4-ball.

Adam Levine

Image description: Figure 1: a cell complex built from two rectangular pyramids by identifying their rectangular faces, and for each short edge of the resulting rectangle, identifying the two faces that border it by folding across the short edge. The result is a cell complex that is almost a thickened 2-sphere except with one unthickened point (the vertex that was the point of the pyramids). Figure 2: a triangulation of the two sphere by two triangles. Caption: Given two triangulated closed 3-manifolds, M1 and M2, with t1 and t2 tetrahedra respectively, we can triangulate the connect sum of M1 and M2 with at most t1 + t2 + 4 tetrahedra by adding in the cell complex in Figure 1. This bound is sharp. Pick a face of M1 and remove a neighbourhood of its interior, like puffing up a paper bag. You get a 2-sphere boundary component, triangulated as in Figure 2. The cell complex in Figure 1 has boundary 2-sphere wedge 2-sphere. Glue one 2-sphere to the new boundary of M1. Repeat for M2.

Image description: a tetrahedron that has been divided into eight pieces by cutting along four triangles and three quadrilaterals. Caption: In 2003, Jaco and Rubinstein introduced the operation of crushing a normal surface, which has since led to remarkably efficient algorithms for unknot recognition, 3-sphere recognition and connected-sum decomposition. Let T be a triangulation of a 3-manifold. A normal surface in T is an embedded surface that meets each tetrahedron of T in a disjoint union of triangular or quadrilateral discs. Pictured on the left is a tetrahedron that has been cut open along a normal surface S. After cutting the entire triangulation in this way, we crush S by collapsing S down to a point. This turns T into a 3-dimensional complex with many non-tetrahedron pieces. For example, in the picture: pieces 1, 2, 7 and 8 become "3-sided footballs"; pieces 4 and 5 become "4-sided footballs"; and pieces 3 and 6 become "triangular purses". We recover a new triangulation T' (that has fewer tetrahedra than T, which is partly why the crushing operation is useful) by flattening the footballs down to edges, and flattening the purses down to triangular faces. For more details, see "A new approach to crushing 3-manifold triangulations" by Ben Burton.

Image description: (a)-(c) a sequence of images depicting a homotopy of a curve eta in the complement of a ribbon disk represented as a surface in S3, with some portions tinted red; (d) two link diagrams, each with two components K and eta, related by isotopy; one includes a ribbon disk for K; (e) a knot diagram; (f) a link diagram with two components K and eta.

Title: A construction of topologically slice knots

Caption: Cochran, Friedl, and Teichner showed that given a (topologically) slice knot K with slice disk Sigma, and an unknot eta in S3 minus K which is nullhomotopic in D4 minus Sigma, then you can ``tie up'' any knot J into K at eta (i.e., infect J by K), and still have a slice knot. For example, say K is a ribbon knot, and eta is as depicted in (a) (the parts of the ribbon disk that are ``deeper'' in B4 are tinted red). Then, we can follow the homotopy depicted in (a)-(c) to see that eta is nullhomotopic in D4 minus Sigma. In particular, this includes the specific example shown in (d), which is known as the ``(positive) Whitehead doubling" operator. The Whitehead double of the (right-handed) trefoil is shown in (e). All Whitehead doubles were already known to be slice by previous work of Freedman, who showed that all knots with trivial Alexander polynomial are slice. But this perspective works in more situations, such as the one in (f), which easily produces knots with nontrivial Alexander polynomial. It is important that we are working in the topological category, as the knot in (e) is not smoothly slice!

For more information, see Cochran, Friedl, and Teichner's paper, New constructions of slice links.

Alex Manchester, Rice University

Image description: A "visual" representation the well-ordering of volumes of hyperbolic 3-manifold with order type ω^ω as well as certain examples of manifolds in the well order.

Caption: This diagram approximately represents the well-ordering on the space of all finite volume hyperbolic manifolds. It turns that the image of the volume function on the space of finite volume hyperbolic 3-manifolds has order type ω^ω. Contrast this to the case of hyperbolic 2-manifolds, where due to Gauss-Bonnet theorem, the possible volumes are multiples of 2π. This is actually true for all hyperbolic n-manifolds by Wang’s Finiteness Theorem! However the case of hyperbolic 3-manifolds is significantly more complicated, given by Thurston-Jørgensen theorem. Interestingly, we can realize many of these manifolds, the smallest volumes are given by arithmetic hyperbolic 3-manifolds and one can produce the limit ordinals via Thurston's hyperbolic Dehn surgery theorem!

For more, check out "Hyperbolic Manifolds" - Marden and

"An Introduction to Geometric Topology" - Martelli

Image : Heawood graph with some limit groups marked

{The Chabauty topology is a topology on the space of all closed subgroups of a group. We can study limits of subgroups in this topology, often they correspond to going to infinity in a symmetric space, or in the $SL_n (\QQ_p)$ analog, to infinity in a building.

Drawn here is the link in $SL_3(\QQ_2)$ for a vertex in the spherical building at infinity. The Heawood graph is also the building for $SL_3(\mathbb{F}_2)$. }

{There are different colors for the vertices, depending on how we go to infinity. The limits of the diagonal group are contained in the groups which stabilize vertices or edges at infinity.

See \href{https://arxiv.org/abs/1711.04864}{https://arxiv.org/abs/1711.04864}. }

Image description: Kirby diagram of the Mazur manifold, consisting of a 2-component link with unknotted components. Caption: Giffen showed that the Mazur manifold M contains a disc D whose complement is homeomorphic to the product of the boundary with the half open interval. Therefore the quotient of M by D is homeomorphic to the cone on the boundary. Bryant showed that if any manifold N contains a disc D in its interior, then N/D x R is homeomorphic to N x R. The only potential non manifold points in the double suspension of the boundary of M lie on the suspension circle. But these have neighbourhoods homeomorphic to cone(boundary M) x R. Therefore we have established Edwards’ result that the double suspension of the boundary of M is the 5-sphere. More generally, Cannon showed that the double suspension of every homology 3-sphere is the 5-sphere. 

Image description: Two Seifert surfaces for a knot J_n

Caption: Here are two genus 1 Seifert surfaces Σ_1, Σ_2 of a knot J_n inspired by [Alford: ’Complements of minimal spanning surfaces of knots are not unique’, Ann. of Math. (2) 91 (1970)]. The boxes denote a zero framed satellite with pattern J and 2n + 1 half-twists respectively. The surfaces are not isotopic in S^3, fixing the knot J_n. Clayton McDonald and I investigated what happens if you allow Σ_1, Σ_2 to move into the 4-ball.

Image description:

Top left: a sutured handlebody. Each row underneath, left: a disc decomposition with two pairs of curves in red and blue, right: a pair of train tracks in red and blue.

Caption:

Consider the sutured handlebody given by the outside of top left. This postcard illustrates some disc decompositions of it. Each decomposition determines a depth 1 foliation, or equivalently an endperiodic surface map. An effort was made to draw each endperiodic map to the right of its respective disc decomposition: Each figure is a union $W^+ \cup K \cup W^-$, where $W^+ \cup K$ contains the blue train track and $K \cup W^-$ contains the red train track. A map $K \cup W^- \to W^+ \cup K$ is determined by the labels on the boundary of these subsurfaces. This extends to an endperiodic map defined on $\bigcup_{i=0}^{\infty} W^+ \cup K \cup \bigcup_{i=-\infty}^{0} W^-$.

Now, an endperiodic map determines a pair of Handel-Miller laminations, see ``Endperiodic Automorphisms of Surfaces and Foliations'', Cantwell-Conlon-Fenley. Train tracks carrying these laminations are drawn on the respective surfaces. By endperiodicity, these induce train tracks on the boundary of the original sutured handlebody. For each of these, the recurrent sub-train track is recorded back on the left as a pair of multi-curves. It is interesting (to me) to study how these curves vary as one moves between different depth one foliations. 

Image description:

Top: Dynkin diagram of A2 and the symmetry planes of the corresponding root system.

Bottom: Dynkin diagram of A2-affine and the corresponding symmetry planes of the root system.

Text: It is possible to classify finite-dimensional simple Lie algebras by their Cartan matrices, and one can associate each such matrix with a diagram (colored graph), called Dynkin diagram. A2 is the Dynkin diagram for the Lie algebra g =sl_3(C).

If we now generalize the matrix, we can reconstruct not only finite-dimensional Lie algebras, but also other types of Lie algebras. One type are affine Kac-Moody algebras (also called affine infinite-dimensional Lie algebras), for example A2-affine.

We obtain the corresponding Dynkin diagram by adding a node to A2.

It is interesting to note that the symmetry group of the root system of A2 is finite, but for A2-affine it is infinite. The pictures below the diagrams show the planes of symmetry.

See "Victor Kac - Infinite dimensional Lie algebras" for more. 

Can you do geometric topology with quantum invariants?

True for twisted Alexander polynomials (non-quantum): they are orders of homology groups which can be computed via Fox calculus

\begin{align*}

\frac{\p (uv)}{\p x}=\frac{\p u}{\p x}\cdot 1+ u\cdot\frac{\p v}{\p x}.

\end{align*}

Now, quantum invariants come from Hopf algebras which typically have elements E,K satisfying $\De(K)=K\ot K$ and

\begin{align*}

\De(E)=E\ot 1+ K\ot E.

\end{align*}

So Fox calculus is a Hopf algebra thing!

This suggests: There is geometric topology in "G-twisted" quantum invariants from (non-semisimple) G-graded Hopf algebras.

Image description:

A genus 2 surface with boundary maps onto the 2-rose. A top-down view of the surface shows the boundary corresponds to the word $[a,b]^3$ in the fundamental group of the rose.

Caption:

In 1981 Marc Culler showed how to study the commutator subgroup of a free group, [F_𝑛,F_𝑛], by mapping surfaces with boundary to graphs. The boundary of a genus 𝑔 surface represents a product of 𝑔 commutators in the image 𝜋_1(Σ)→𝜋_1(Γ), and the converse holds as well: any product of commutators gives rise to a map from a surface to a graph.

The picture exhibits the well-known fact that in [F_2,F_2 ], the element [𝑎,𝑏]^3 can be expressed as a product of only two commutators. It also forms the first step in an algorithm to determine what those commutators are by cutting up the surface into discs: [𝑎𝑏𝑎^{−1}, 𝑏^{−1} 𝑎𝑏𝑎^{−2}][𝑏^{−1} 𝑎𝑏, 𝑏^2].

Image description: a multicolored set of points in the complex plane trace the outline of a "hairy" circle. Caption: Pseudo-Anosovs play a central role in the theory of mapping class groups of surfaces. Certain interval maps can be realized as the train track map for a pseudo-Anosov. On the left is the locus of Galois conjugates of stretch factors for pseudo-Anosovs whose train track maps may be realized by a special type of interval map with two critical points. Of note is the fact that this set is highly concentrated about the unit circle, with “parentheses” or “hairs” branching off.

For more, see “Constructing pseudo-Anosovs from expanding interval maps,” by F. The image on the left is to appear in a forthcoming sequel by the same author.

Image description: A doubly-pointed Heegaard diagram $(T^2,\alpha_1,\beta_1,z,w)$ of a constrained knot $K$ in the lens space $L(5,2)$.

Caption: This figure shows the diagram to define a constrained knot. The Heegaard diagram $(T^2,\alpha_0,\beta_0)$ illustrates a standard Heegaard splitting of the lens space $L(5,2)$. The curve $\alpha_1$ is the same as $\alpha_0$ and the curve $\beta_1$ is chosen to be disjoint from $\beta_0$. Two basepoints $z$ and $w$ indicate how to contruct the knot $K=a\cup b$: choose an arc $a$ (resp. $b$) in $T^2-\alpha_1$ (resp. $T^2-\beta_1$) connecting $z$ to $w$ and push it in the handlebody corresponding to $\alpha_1$ (resp. $\beta_1)$. Knot Floer homology and instanton knot homology of constrained knots are isomorphic. For more, see "Constrained knots in lens spaces", Fan Ye 2020; "Instanton Floer homology, sutures, and Heegaard diagrams", Zhenkun Li and Fan Ye 2020; "Instanton Floer homology, sutures, and Euler characteristics", Zhenkun Li and Fan Ye 2021.

Iterating Pappus' theorem produces often fractal "limit curves" (three examples are pictured, approximately, together with the configurations of lines and points producing them), and, related representations of PSL(2,Z) into PGL(3,R) with interesting properties. Reference: Rich Schwartz, “Pappus’s theorem and the modular group”.

Top Left Text: Greetings from T(R)!

Image: Four images of genus two surfaces connected by arrows. Each surface displays an orange simple closed curve whose length shrinks in each successive surface. In the fourth surface the curve breaks into two punctures.

Bottom Text: One way to move to the boundary of Teichmüller space is by pinching a curve so that it “breaks” the surface. With the Teichmüller metric this path has infinite length, but in the Weil-Petersson metric it is finite, so the space is incomplete. These metrics fit into the family of 𝑳^𝒑 metrics on Teichmüller space, the Weil-Petersson metric is the 𝑳^𝟐 metric and the Teichmüller metric is the 𝑳^∞metric. In each of the 𝑳^𝒑 metrics this path has finite length, but that length goes to infinity with 𝒑!

Image description:

A distorted grid of purple cells, with a red blob in each cell.

Caption:

A view from within the mapping torus with monodromy ((1,1),(0,1)). We generated this image by simulating light rays travelling along geodesics in Nil geometry, the Thurston geometry for this manifold.

For more, see "Ray-marching Thurston geometries", arXiv:2010.15801

Remi Coulon, CNRS

Sabetta Matsumoto, Georgia Tech

Henry Segerman, Oklahoma State

Steve Trettel, Stanford

Image description: two slice disks drawn with many curvy, black stripes on a postcard trimmed with many purple, blue, and pink rhombi. Caption: This postcard illustrates two slice disks bounding the same prime knot K, drawn here in the 3-sphere. In fact, there are two more slice disks for K that I did not draw. These four slice disks have a pretty neat property: they are all unique up to smooth, boundary-preserving isotopy through the 4-ball. To prove this, we show that the maps that they induce on Khovanov homology are all distinct (which is enough to guarantee that they are distinct - more details in my talk!). In fact, for each positive integer n we can come up with a similar knot that has 2^n unique slices, distinguished by their induced maps on Khovanov homology!

Image 1 description: some linked circles, one of which is red and labeled as F, each labeled by an integer.

Image 1 caption: This surgery on F, a regular fiber of the Poincaré homology sphere, is the connected sum of lens spaces L(2,1)#L(3,2)#L(5,4). These two diagrams are related by a slam-dunk in Kirby calculus.

Image 2 description: many linked circles, one of which is green and labeled as E and one of which is blue and labeled as C*, each labeled by an integer.

Image 2 caption: These two surgery diagrams present knots E and C in P with E(7/2) = L(7,6) and C(27/2) = L(27,16).

Body text:

Dehn surgery is a cut-and-paste construction that plays a fundamental role in the study of 3-manifold topology. Every closed 3-manifold may be obtained by Dehn surgery on a link in any other 3-manifold---this is similar to the fact that in every closed 3-manifold there is a surface that cuts the 3-manifold into two handlebodies. What if we focus on what can happen when we perform Dehn surgery on a single knot?

The fact that any two link surgery diagrams presenting the same 3-manifold are related by a finite sequence of well understood diagrammatic moves makes coming up with examples of interesting surgeries on knots a fun exercise. An even more interesting task, with more significant topological consequences, is to show that these interesting surgeries are unique in some way.

For example, if a knot in the Poincaré homology sphere admits a surgery to a connected sum of lens spaces with at least three summands, then it is isotopic to the red knot in Figure 1. If half-integer surgery on a knot in the Poincaré homology sphere is a lens space, then the lens space is one of the two lens spaces described in Figure 2.

For more on this perspective on Dehn surgery (which is one of many!), see "Three lens space summands from the Poincaré homology sphere", C.

Jacob Caudell, Boston College

Image description: A hyperbolic triangle;

The text: Searching for obstructions to the existence of

a Riemannian metric with positive scalar curvature on

a closed non-simply connected smooth manifold,

we find that the connect sum of a closed n-manifold

with David’s exotic aspherical n-manifolds (n <9)

does not admit a Riemannian metric with

positive scalar curvature.

For more, see Enlargeable Length-structures and

Scalar Curvatures, D. 2021, Ann. Global Anal. Geom. 

Image description: An immersed curve in the plane minus the lattice representing knot Floer homology of T(2,5) and a line of slope 1, along with the complex computed from this pair of curves, the knot Floer homology of the dual knot in +1 surgery on T(2,5).

Caption: Given a knot $K$ in $Y^3$ and $p/q$ in $\Q$, the Heegaard Floer homology of the surgery $Y_{p/q}(K)$ can be computed using the mapping cone formula of Ozsváth and Szabó. This was enhanced by Hedden and Levine to recover the knot Floer homology of the dual knot $K^*$ (the core of the filling torus) as well. While tremendously useful, the surgery formula is a bit unwieldy algebraically. Fortunately, it admits a nice geometric description. The knot Floer complex $CFK(Y,K)$ can be interpreted as an immersed multicurve in the torus with one marked point. To compute the $\F[U,V]$ complex $CFK(Y_{p/q}(K), K^*)$, we take Floer homology with a line of slope $p/q$ through the marked point. The figure shows the case that $K = T(2,5)$ in $S^3$ and $p/q = 1$. For clarity, the curves are lifted to the plane. There are 7 generators, with the Alexander grading increasing each time the marked point is passed moving along the blue line. We place basepoints $z$ and $w$ next to the marked points, and the differential counts bigons (shaded), weighted by $U$ (resp. $V$) for each $w$ (resp. $z$) covered.

Image description: A rectangle joined to two squares, with edges divided into smaller segments whose lengths form a geometric sequence.

Text: In this figure, each segment labeled A_n, B_n, C'_n, or C''_n has length 1/2^n. The endpoints of the segments are removed, and each pair of edges with the same label is identified by translation in R^2, as is the remaining pair of unlabeled edges on the squares S_1 and S_2. The resulting surface has infinite genus and one topological end. This surface admits an affine homeomorphism psi that is pseudo-Anosov. Expand the central rectangle R horizontally by a factor of 2, and contract R vertically by a factor of 1/2 to obtain psi(R). Do the same with the rectangle R' which is the union of S_1 and S_2 (the top edge of S_1 is glued to the bottom edge of S_2) to obtain psi(R'). Take psi(R) and lay it over S_1 and the lower half of R, and lay psi(R') over S_2 and the top half of R. This affine map is compatible with all identifications.

From J. Bowman, "The complete family of Arnoux–Yoccoz surfaces" (2012).

Joshua Bowman, Pepperdine University

Image description: A Jordan curve. A Klein bottle. A shrinking wedge of circles. A sphere eversion. Each math object has two human figures nearby. A stamp featuring Henri Poincaré. Caption: The topology of outreach. What does it mean to be “in” math? To: You. The very idea of outreach implies that one can be “in” or “out” of math. From an early age and throughout our lives we have experiences that tell us whether we have a place in math, or not. Sometimes the message is complex; sometimes it is painfully blunt. Sometimes we can go very far, but since we never pass through a certain boundary, we do not think we have made it. Sometimes there are many layers and hurdles—too many—and sometimes the path is strikingly smoother. But all told, if we plan and work together, we can build a math world where the hard distinction between “in” and “out” is erased—where everyone can find a place in math. From: Justin Lanier.

Image: A knot with five twisting regions, where the number of crossings in each is labelled.

Caption: The smallest "even" Montesinos knot such that the manifolds obtained by ±1-surgery cannot be distinguished by either Heegaard Floer methods as in [1], or finite type invariants from [2], or even the Reshetikhin-Turaev invariant techniques from [3].

(Don't worry: this is not a counterexample to the Cosmetic Surgery Conjecture as the two manifolds apparently have different hyperbolic volumes according to SnapPy).

[1] J. Hanselman, "Heegaard Floer homology and cosmetic surgeries in S3"

[2] T. Ito, "On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in S3"

[3] R. Detcherry, "A quantum obstruction to purely cosmetic surgeries"

Image description: three tangles contained in diamonds and one much larger tangle built from copies of the first three. Caption: "These are tileable tangle diagrams for common crochet stitches"

Image description: An octopus portrayed as an infinite-type surface with finitely many ends accumulated by genus drawn in white, with a multicurve drawn in bright pink and purple in the genus 4 compact core (the head of the octopus).The background is a dark navy blue. Caption: An end-periodic he'e. End-periodic homeomorphisms arise as monodromies of the non-compact leaves of depth-one foliations of 3-manifolds.

Image description: A blue knot and a red red knot joined by an orange band.

Caption: A split link consisting of a T(2,3) torus knot (a.k.a. the trefoil) and an unknot.Performing a band sum along the band indicated in the figure yields the knot 8_11.Thinking of the unknot and the band as a birth and a saddle, respectively, this picture can be interpreted as showing a ribbon concordance from T(2,3) to 8_11, both of which happen to be 2-bridge knots.In fact, any ribbon concordance from a 2-bridge knot to another must "start" at a 2-bridge knot of the form T(2,n), for some n. This is shown in the paper "Ribbon cobordisms between lens spaces".

Two facts about exotic manifolds: 1) Exotic 4-manifolds don't have to be complicated. RP^4 \# K3 is homeomorphic but not diffeomorphic to

RP^4 \#^{11} S^2 x S^2.

2) The smooth generalised Poincar\'{e} conjecture is open in infinitely many dimensions. It is true in dimensions 1, 2, 3, 5, 6, 12, 56, and 61.  It is false in all other odd dimensions \geq 7, and all other even dimensions 8 \leq n \leq 124.   126 and 4 are the most interesting open dimensions.

Image description: A sequence of four labeled graphs of groups. The first graph of groups has an unlabeled center vertex and seven edges. Four of the edges form loops labeled x, y, z and w. The remaining three edges connect to valence-one vertices labeled ⟨a⟩, ⟨b⟩ and ⟨c⟩. The second graph of groups is obtained from the first by collapsing the edge incident to the vertex ⟨a⟩. Thus the new graph of groups has center vertex labeled ⟨a⟩. An initial segment of the edges labeled x and z, as well as a terminal segment of the edges labeled y and z, and a segment of the edge connecting the center vertex to ⟨b⟩ are all colored red. The third graph of groups is combinatorially identical to the second, but the labels now read ax, ya^-1, aza^-1 and w for the loop edges, and ⟨aba^-1⟩ and ⟨c⟩ for the valence-one vertices. The fourth graph of groups is obtained from the third by expanding so that ⟨a⟩ is yet again a valence-one vertex; so the fourth graph of groups is combinatorially identical to the first, but the labels match the third graph of groups instead.

Caption: This postcard depicts a *collapse Whitehead move* in the spine of Guirardel–Levitt’s Outer Space for a free product, in this case the free product F of three finite cyclic groups ⟨a⟩, ⟨b⟩ and ⟨c⟩ with a free group of rank 4, ⟨x, y, z, w⟩. The process, reading left to right, top to bottom, is as follows: collapse the edge incident to ⟨a⟩, twist by an automorphism, then expand so ⟨a⟩ is again a valence-one vertex.

Let me explain. The spine of Guirardel–Levitt Outer Space is a simplicial complex whose vertices represent certain actions of F on trees. Edges in Outer Space correspond to equivariantly collapsing or expanding certain families of edges in the trees. What’s pictured are four different quotient *graphs of groups* labeled to describe an isomorphism of their fundamental group with F. Two of the graphs of groups, the upper right and lower left, represent the same vertex of the spine of Outer Space. This is because they differ by an automorphism of graphs of groups which “twists” the oriented edges labeled in red by a. This twisting cannot be realized as an automorphism of the graphs of groups on the upper left or lower right, so the process describes a path of length two in Outer Space.

Image: bordered Floer invariants: CFA hat of the torus knot -T2,7 and CFD hat of Q(-T2,7) where Q is the Mazur pattern. Caption: A satellite operator, or pattern P, is a knot inside S1 x D2. Patterns act on knots via the satellite construction. Any satellite operator P gives a function on the smooth concordance group. For nearly all satellite operations, there is no obvious way to compute CFK \infty (P(K)) from CFK \infty (K). However, passing to bordered Floer homology, we can. In addition to giving us new families of knots to study, we can study the images of satellite operators in C to better understand its group structure. 

Image description: On the left hand side is a schematic representing a (g,k)-trisection of a 4-manifold X. This is a circle divided into three pieces. The center point represents a genus g surface, the spokes represent three genus g handlebodies, the union of two spokes represents a connect sum of k copies of $S^1 \times S^2$, and a sector represents a boundary connect sum of k copies of $S^1 \times B^3$. On the right hand side is a cube of fundamental groups of each of the pieces of the trisection. The fundamental group of the surface maps to the fundamental groups of the three handlebodies. These in turn map to the fundamental groups of the three sectors, which then all map to the fundamental group of X. An arrow goes from left to right reading "Van Kampen Theorem," and another arrow goes from right to left reading "Abrams, Gay, Kirby." Caption: Group Trisections. Sarah Blackwell. (g,k)-group trisection of a group G [Abrams, Gay, Kirby] = commutative cube of groups (as shown below) such that every homomorphism is surjective and each face is a pushout. 

Image description: Several two and three dimensional examples of canceling handles. The two dimensional examples show a 0 and 1 handle canceling pair and a 1 and 2 handle canceling pair. The three dimensional examples show a 1 and 2 handle canceling pair, a 0 and 1 handle canceling pair, and a 2 and 3 handle canceling pair. Caption: Building a Bridge (refers to adding a three dimensional 1 handle) = Digging a Tunnel (refers to removing a three dimensional 2 handle). Sarah Blackwell. Some examples of canceling handles in a handle decomposition. "What are the real-world applications?" they asked...

Image description: On the left hand side is a tic-tac-toe board, with an X in the second column of the first row, first column of the second row, and third column of the third row. The X in the third column of the third row is highlighted. Another highlighted X lies to the left of the diagram, aligned with the third row. A diagonal line is drawn connecting this X with the other non-highlighted X's. On the right hand side is a schematic showing how to construct a model of torus tic-tac-toe, that starts with a tic-tac-toe grid drawn on a square, and shows how joining up the opposite sides of a square results in a torus. Caption: Torus Tic-Tac-Toe: an outreach activity. Sarah Blackwell. Torus tic-tac-toe is tic-tac-toe played on a torus, so the X's above are now in a winning position. (The highlighted X's are the same.) Instructions for creating your own game, from a lesson plan by me and Devashi Gulati. Inspired by "The Shape of Space" by Jeffrey Weeks.

Image description: A colourful, densely squiggling pair of curves, on a black background.

Caption: Cannon and Thurston proved that, given a closed hyperbolic three-manifold $M$ fibering over the circle with fibre $F$, there is an associated sphere-filling curve.

That is, there is a continuous, surjective, $\pi_1(M)$-equivariant map from the Gromov boundary of $\pi_1(F)$ to that of $\pi_1(M)$. This is the first example of a Peano curve appearing ``naturally'' in mathematics.

Using tools from theory of veering triangulations we produce new examples which, furthermore, do not come from surface subgroups. Pictured here is a fundamental domain, under the parabolic subgroup, of an approximation to the image of the veering circle for the snappy manifold s227 inside of the Bowditch boundary of $\pi_1(M)$.

Jason Manning, Cornell University

Saul Schleimer, University of Warwick

Henry Segerman, Oklahoma State University

Image description: A fractal like the serpinski carpet, but skewed. Text:   A part of the limit set of the mapping class group of a non-orientable surface of genus $6$ when restricted to a $3$-dimensional stratum of projective measured foliations.

  Unlike in the orientable case, the limit set is not the entire stratum, but rather a nowhere dense measure $0$ subset.

  The Hausdorff dimension of the limit set restricted to a dimension $n$-stratum is bounded between $n-1$ and $n-2$, where the upper bound is sharp.

  Locally, the limit set is contained in a hyperplane, and the image is a rendering of the limit set when restricted to one such hyperplane.

Image Description: A grid of circles with a pair of numbers inside each one. Green edges connect the circles vertically and dark blue in the horizontal direction. Two families of nested ellipses are behind the grid, one in blue in the upper right, another magenta one more centered. Each is done using some transparency so that the smaller ellipses are darker and the overlap has a blend of the colors. The ellipses pass through the circles and mark superlevel sets for the numbers in the circles.

Caption: Knot lattice homology is defined using two filtrations (where one is a translation of the other) on a cube decomposition of euclidian space.

Image description: a coloured circle on the left, and four coloured points on the right, with certain indicated groupings representing the basis of a topology on the four points. Caption: The map from the circle to the four-point space (giving by mapping each colour to itself) is a weak homotopy equivalence.

Caption: This postcard is on Ratner's orbit closure theorem - we begin with an illustrative example to which it applies. Here's the setup: let $[x]$ denote a point in the $n$-torus $\TT^n = \RR^n / \ZZ^n$, i.e. $[x] = x + \ZZ^n$ is an equivalence class of vectors in $\RR^n$. Fix any vector $v \in \RR^n$. The vector $v \in \RR^n$ now determines a smooth flow on the torus by the map

$$\varphi_t ([x]) = [x + tv] \text{ where } x \in \RR^n \text{ and } t \in \RR.$$

It can be proved that the closure of the orbit of each point $x \in \TT^n$ is a subtorus of $\TT^n$. More precisely, there is a vector subspace $S \subseteq \RR^n$ such that

$v \in S$ (so the entire $\varphi_t$ of $[x]$ is contained in $[x+S]$),

 the image $[x+S]$ of $x+S$ in $\TT^n$ is compact (hence, the image is diffeomorphic to $\TT^k$ for some $k \in \{0,1,2,\dots,n\}$), and 

the $\varphi_t$-orbit of $[x]$ is dense in $[x+S]$ (so $[x+S]$ is the closure of the orbit of $[x]$).

In short, the closure of every orbit is a nice, geometric subset of $\TT^n$.

Ratner's Orbit Closure Theorem is a vast generalization of the above example. It allows 

the Euclidean space $\RR^n$ to be replaced by any Lie group $G$,

the subgroup $\ZZ^n$ to be replaced by any lattice $\Gamma$ in $G$, and

the map $t \mapsto tv$ to be replaced by any unipotent one-parameter subgroup $u^t$ of $G$. (``unipotent" means all eigenvalues are equal to 1).

Image description: A 2-torus with a periodic, closed orbit of a point drawn on it.

This postcard demonstrates an example of how a torus knot T(p,q) can be viewed as braided around a torus in two different ways, one with p strands and one with q strands. The postcard is "sent" from Siddhi to her parents to show them what her REU students are currently learning about. 

Image description: surgery description of a closed 3-manifold. Caption: This is a hyperbolic rational homology 3-sphere that has only a finite number of (P)SL(2,C) representations. This manifold, constructed by Boyer-Zhang, contains one embedded surface which cannot be seen by ideal points of curves in the SL(2,C)-character variety (because there are no curves). However, it is known (using a theorem of Friedl-Kitayama-Nagel) that the embedded surface in this 3-manifold corresponds to an ideal point of a curve in the SL(2d,C)-character variety of the manifold for some positive integer d.

Image description:  On the left is a triangular tessellation of the Poincare disk. Each triangle has a numbered circle inside it, with numbers from 1-14. There are red, blue, and black arcs between the circles, making a tree. On the right is a directed graph with vertices numbered 1-14 and red, blue, and black edges labelled with "a", "b", and "c."

Image description: a surface of genus one with two boundary components \delta_1 and \delta_2, a red meridian \alpha, a red longitude \beta, a blue curve around both boundary components \gamma, a red boundary-parallel curve about \delta_1, 4+n red boundary-parallel curves about \delta_2.

This somewhat suspicious \Sigma_{1,2} illustrates an infinite family of open books with monodromies \varphi_n = \tau_\alpha \tau_\beta \tau_\gamma^{-1} \tau_{\delta_1} \tau_{\delta_2}^{4+n} for n \geqslant 0 . With Wand, we have shown that for all n , they support Stein-fillable contact manifolds, but \varphi_n does not admit a factorisation into a product of positive Dehn twists; this makes them first known such examples of genus one. Together with previous work of Wand and Wendl, this proves that a correspondence between Stein fillings and positive factorisations of monodromies only exists for planar open books. See more in our preprint on the arXiv: \href{https://arxiv.org/abs/2103.13250}{2103.13250}.

Image description: a finite maximal geodesic lamination on a closed surface of genus g=2 consisting of a collection of 2 non-homotopic disjoint simple closed curves (blue), and 6 bi-infinite geodesics (pink, brown, dark green, red, orange, light green) whose ends spiral around these closed geodesics and which split the surface into 4 infinite triangles (yellow, white).

Caption: The picture below shows a finite maximal geodesic lamination on a closed surface S of genus g=2. It consists of a collection of s non-homotopic disjoint simple closed curves (blue) with 1≤s≤3g-3, and 6g-6 bi-infinite geodesics (pink, brown, dark green, red, orange, light green) whose ends spiral around these closed geodesics and which split the surface S into 4g-4 infinite triangles (yellow, white).

To this data one can associate Thurston’s shearing coordinates for the Teichmüller space T(S). This can be generalized to define the Bonahon-Dreyer coordinates for the Hitchin component: a distinguished component in the space Hom(\pi_1(S), PSL(n,R))/PSL(n,R) that contains a representation of the form \iota_n \circ \rho, where \rho \in T(S) and \iota_n  is the unique irreducible n-dimensional representation of PSL(2,R). For more information see Parameterizing Hitchin Components by Bonahon-Dreyer.

Image description: the orbit of a circle under a Kleinian group drawn on a sphere, which limits to the orbit of anther circle. Caption: An isometrically immersed totally geodesic hyperbolic plane in a complete hyperbolic 3-manifold of finite volume is either closed or dense, independently due to Ratner and Shah. Recently, McMullen, Mohammadi and Oh generalized this dichotomy to portions of geodesic planes in the convex core of any convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume. A natural question arises: if a plane is closed in the convex core, is it closed in the whole manifold? The answer is no, and the figure on the left exhibits such a counterexample. For details, see [Y. Zhang, "An exotic plane in an acylindrical 3-manifold"].