Schedule of Topology Seminars

I'll talk about smooth embeddings of closed 3-manifolds into 4-space. This talk can be regarded as part 2 of a lecture I gave in this seminar last Spring, but will be self-contained. The topic has much to do with the major problems of low dimensional topology: the Schoenflies conjecture, the smooth 4D Poincare conjecture, and the (negation of the) Andrews-Curtis conjecture.

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the Cartan subgroup in SL(n,R). For n=3, the Cartan subgroup has precisely 5 limits, and for n=4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n > 6, there is a continuum of non-conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

In this talk, we'll be concerned with producing reducible 3-manifolds via Dehn surgery on a knot in the three-sphere. The proposed classification of such surgeries is due to Gonzalez-Acuna and Short, known as the Cabling Conjecture, which asserts that only a certain surgery on cable and torus knots (and 0-surgery on the unknot) can give such manifolds. Using these surgeries as a guide, a large amount of progress toward the conjecture has been established which essentially tells you that an arbitrary reducible Dehn surgery is coarsely similar to the cabled reducible surgery. In a similar spirit, it should be the case that all reducible Dehn surgeries on nontrivial knots give precisely two irreducible connected summands, sometimes referred to as the Two Summands Conjecture. Using the main combinatorial object appearing first in the proof of the Knot Complement Problem due to Gordon and Luecke, we are able to restrict any surgery coefficient producing more than two summands to being less than or equal to the bridge number of the purported knot. A consequence of this is the completion of the Two Summands Conjecture for positive braid closures and knots with bridge number less than or equal to five, and another proof of the Cabling Conjecture for knots with bridge number less than or equal to 3.

This introductory talk is aimed at graduate students and will give an overview. Given two groups G and H the direct sum G x H is a group structure on the cartesian product. But there is also the notion of the semi-direct product of two groups, which is a twisted product. In topology a product bundle is given by a projection p : B x F —> B. A bundle is a map that is locally like this. The simplest example is a Mobius band where B is a circle and F is an interval. The tangent bundle to a smooth manifold B is the set of all tangent vectors for B. In this case F is Euclidean space. The Hairy Ball theorem shows that the tangent bundle for a sphere is not a product bundle. I might try to describe how one classifies these objects and how one measures the different ways bundles twist using characteristic classes, connections and so on. I might talk about circle bundles and Seifert fiber spaces. Then again, I might not.

Recent advancements in the study of high-distance surfaces in knot exteriors have made it possible to construct knots in a way that puts heavy restrictions on all essential and strongly irreducible surfaces in the knot exterior. We use these techniques to construct the first examples of knots in Gabai thin position which admit compressible thin levels. This is joint work with Alex Zupan.

I shall introduce certain deformations of geometric structures on a surface that yield holomorphic disks in its Teichmüller space, and discuss their asymptoticity in the Teichmüller metric. The proofs consider certain degenerate surfaces that arise as geometric limits, and involve constructions of quasiconformal maps with low quasiconformal distortion.

The study of Left-Orderable groups has deep connections to geometry, topology and dynamics. A well known and very useful tool is the fact that all countable L.O. groups embed in the group of homeomorphisms of the line. Uncountable L.O. groups act on ordered spaces too, but whether they act on the line is a complicated question. In this talk, I'll present a natural -- though perhaps surprising -- example of a L.O. group that does not act on the line. The proof is entirely self-contained, using one sophisticated fact (involving bounded cohomology) but otherwise a very hands-on approach.

In this talk we consider the SL(2,C)-character variety of the four-holed sphere S, and the natural action of the mapping class group MCG(S) on it. In particular, we describe a domain of discontinuity for the action of MCG(S) on the relative character varieties, which is the set of representations for which the traces of the boundary curves are fixed. Time permitting, in the case of real characters, we show that this domain of discontinuity may be non-empty on the components where the relative Euler class is non-maximal. I will focus on the combinatorial methods of the proofs. (This is joint work with F. Palesi and S. P. Tan.)

A shaped triangulation is a finite triangulation of an oriented pseudo three manifold where each tetrahedron is an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3-2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges. Similarly to Turaev-Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions. This is a joint work with R. Kashaev and G. Vartanov.

Kerckhoff's solution to the Nielsen realization problem showed that the action of any finite subgroup of the mapping class group on Teichmüller space has a fixed point. The set of fixed points is a totally geodesic submanifold. We study the coarse geometry of the set of points which have bounded diameter orbits in the Teichmüller metric. We show that each such almost-fixed point is within a uniformly bounded distance of the fixed point set, but that the set of almost-fixed points is not quasiconvex. In addition, the orbit of any point under the action of a finite order mapping class is shown to have a fixed coarse barycenter. In this talk, I will discuss the machinery and ideas used in the proofs of these theorems.

It is an interesting question to determine how complex it is to actually compute topological invariants of certain spaces from a suitaqble algebraic model. For example, it is known that computing the Betti numbers of topological spaces can be NP hard. In this talk I shall approach the following concrete problem of that kind: Given a simply connected space X with both \(H_*(X;\mathbb{Q})\) and \(\pi_*(X)\otimes \mathbb{Q}\) being finite dimensional, what is the computational complexity of an algorithm computing the cup-length and the rational Lusternik-Schnirelmann category of \(X\)? How complicated is it to decide whether a Lie group action on a compact manifold has only finite isotropy, i.e. is almost free? Basically, by a reduction from the decision problem of whether a given graph is \(k\)-colourable for \(k\geq 3\) it will be shown that even stricter versions of these problems are NP-hard.

(Joint with Caglar Uyanik.) Following a strategy developed by Athreya and Cheung, we compute the gap distribution of the slopes of saddle connections on the octagon by translating the problem to a question about return times of the horocycle flow to an appropriate Poincare Section. This same strategy was used by Athreya, Chaika, and Lelievre to compute the gap distribution on the Golden L. The octagon is the first example of this type of computation where the Veech group has two cusps.

In this talk I will survey some of Minkowski's ideas and their modern incarnations.

An essential hypersurface, \(N\), in a manifold, \(M\), gives rise to decomposition of fundamental group as either an amalgamated free product or an HNN extension. In the context of geometric structures this decomposition can give rise to deformations when the fundamental group of the hypersurface has non-trivial centralizer. In this talk we will explore the situation where the \(M\) is hyperbolic with finite volume and \(N\) is totally geodesic. This is Joint work with Ludovic Marquis.

Every subgroup \(G\) of the outer automorphism group of a finite-rank free group \(F\) naturally determines a free group extension \(1\to F \to E_G \to G\to 1\). In this talk, I will discuss geometric conditions on the subgroup \(G\) that imply the corresponding extension \(E_G\) is hyperbolic. Our conditions are in terms of the free factor complex and are related to certain aspects of hyperbolicity in the Culler-Vogtmann Outer space of \(F\). As an application, we construct new examples of hyperbolic free group extensions. Joint with Samuel Taylor.

A real projective structure on a manifold or an orbifold is given by locally modeling the space by pieces of real projective space glued with projective patching maps. Hyperbolic manifolds are examples. In this talk, we will give some survey of results for deformation spaces for closed manifolds and orbifolds. A generalization Weil's work shows that the deformation spaces of convex real projective structures on closed manifolds or orbifolds are usually open and closed in the PGL(n+1, R)-character varieties. We will try to prove the openness and closeness for the deformation spaces of convex real projective structures on orbifolds with ends satisfying some conditions. (Cooper, Long, and Tillman are considering structures with analogous types of ends.)