# Seminars & Events for Topology Seminar

##### Topologically minimal surfaces in 3-manifolds

Topologically minimal surfaces are the topological analogue of geometrically minimal surfaces. Such surfaces generalize well known classes, such as incompressible, strongly irreducible (or weakly incompressible), and critical surfaces. Applications include problems dealing with stabilization, amalgamation, and isotopy of Heegaard splittings and bridge spheres for knots. In this talk we will review the basic definitions and discuss both existing and potential applications of this new theory.

##### Persistence of Essential Surfaces after Dehn filling

We show that the set of closed, essential, 2-sided surfaces (considered up to isotopy) in a 3-manifold with a torus boundary component survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a very constrained way. If time permits, we will also sketch future work on the persistence of the set of Heegaard surfaces after generic Dehn filling. This is joint work with Ryan Derby-Talbot and Eric Sedgwick.

##### Rational homology disks and symplectic topology

Surface singularities admitting a smoothing with the homology of the 4-disk (a so-called rational homology disk, QHD) play central role in the construction of exotic 4-manifolds through the "rational blow down process." Applying various forms of gauge theory we derive obstructions for a singularity to admit such smoothings. Using symplectic topology we classify all starshaped resolution graphs with QHD smoothings.

##### Immersed surfaces in closed hyperbolic 3-manifolds

Given any closed hyperbolic 3-manifold $M$ and $\epsilon > 0$, we find a closed hyperbolic surface $S$ and a map $f\to S\to M$ such that $f$ lifts to a $1+\epsilon$-quasi-isometry from the universal cover of $S$ to the universal cover of $M$. This is joint work with Vladimir Markovic.

##### More on immersed surfaces in closed hyperbolic 3-manifolds

##### Cusp volume of fibered 3-manifolds

Consider a 3-manifold $M$ that fibers over the circle, with fiber a punctured surface $F$. I will explain how the volume of a maximal cusp of $M$ (in the hyperbolic metric) is determined up to a bounded constant by combinatorial properties of the arc complex of the fiber surface $F$. This is joint work with Saul Schleimer.

##### Link surgery, monopole Floer homology, and odd Khovanov homology

I'll describe new bigraded invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction relates the topology of link surgeries to the combinatorics of polytopes called graph associahedra. For a link in the 3-sphere, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, $Z/2Z$ Khovanov homology and a version of the monopole Floer homology of the branched double cover. This perspective also yields a simple, topological proof that odd Khovanov homology is mutation invariant.

Paper references: http://arxiv.org/abs/0903.3746, http://arxiv.org/abs/0909.0816

##### A combinatorial approach to harmonic maps

I will discuss joint work with Hass on a combinatorial approach to harmonic maps. This work is still in progress. Such an approach has been used previously by other authors for computational reasons, but we are developing our approach with an eye to theoretical applications. Some applications in low dimensional topology will be discussed.

##### Comultiplication in link Floer homology and transversely non-simple links

By theorems of Bennequin, Wrinkle and Orevkov-Shevchishin, transverse links in the unique tight contact structure on $R3$ may thought of as closed braids. For a word $h$ in the braid group on $n$ strands, let us denote by $T_h$ the corresponding transverse link. In this talk, I'll describe a relationship, for two braid words $h$ and $g$, between the transverse link invariants in Floer homology associated to $T_g$ and $T_h$ with the invariant associated to $T_{hg}$. And I'll describe how this relationship can be used to produce a plethora of new prime transversely non-simple link types.

##### Knots with small rational genus

If $K$ is a rationally null-homologous knot in a 3-manifold $M$ then there is a compact orientable surface $S$ in the exterior of $K$ whose boundary represents $p[K]$ in $H_1(N(K))$ for some $p > 0$. We define $\Vert K \Vert$, the *rational genus* of $K$, to be the infimum of $-\chi^-(S)/2p$ over all $S$ and $p$. If $M$ is a homology sphere then this is essentially the genus of $K$. By doing surgery on knots in $S3$ one can produce knots in 3-manifolds with arbitrarily small rational genus. We show that such knots can be characterized geometrically.

##### Bundle structures and Algebraic K-theory

This talk will describe (Waldhausen type) algebraic K-theoretic obstructions to lifting fibrations to fiber bundles having compact smooth/topological manifold fibers.The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions.

##### Bordered Floer homology and factoring mapping classes

I will discuss "bordered Floer homology", an invariant for three-manifolds with parameterized boundary. The theory associates a differential graded algebra to a (parameterized) surface; and a module over that algebra to a three-manifold which bounded by the surface. I will describe this construction, and then focus on computational aspects of this theory, including an algorithm for calculating HF-hat of closed three-manifolds, obtained by factoring mapping classes. This is joint work with Robert Lipshitz and Dylan Thurston.

##### Quasigeodesic pseudo-Anosov flows

A quasigeodesic is a curve which is uniformly efficient in measuring distance in relative homotopy classes or equivalently efficient up to a bounded multiplicative distortion in measuring distance when lifted to the universal cover. A flow is quasigeodesic if all flow lines are quasigeodesics. The talk will explore quasigeodesic pseudo-Anosov flows in atoroidal 3-manifolds, of which there are several infinite families of examples. By geometrization and irreducibility the manifolds are hyperbolic. In such manifolds quasigeodesics are extremely important as for instance they are a bounded distance from minimal geodesics (in the universal cover). One important result is that such flows induce ideal maps from the ideal boundary of the stable/unstable leaves to the boundary of hyperbolic 3-space.

##### Grothendieck's problem for 3-manifold groups

##### The proof of the Periodicity Theorem

The Periodicity Theorem is one of the key steps in the proof of the Kervaire Invariant Theorem. Its proof involves methods from equivariant stable homotopy theory including computations with $RO(G)$-graded homotopy groups.

##### Heegaard Floer Homology and Knot Surgeries

Wallace and Lickorish showed that any 3-manifold can be realized as surgery on a link in S3; however, fifty years later, we still have a rather poor understanding of which manifolds can be constructed from surgery on a knot and which knots give these surgeries. For example, the Berge Conjecture attempts to list all knots which can give rise to a lens space. We will ask a slightly easier question, that of which spherical Seifert fibered spaces (aka spherical space forms) arise as knot surgeries. We will use an obstruction from Heegaard Floer theory, the "correction terms" assigned to a manifold and its associated spin^c structures. These terms can be calculated either from a knot surgery description of a manifold or from a description of it as a Seifert fibered space.

##### Cosmetic Surgery Conjecture on S3

It has been known over 40 years that every closed orientable 3-manifold is obtained by surgery on a link in $S3$. However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description. In this talk, we discuss the following uniqueness theorem for Dehn surgey on a nontrivial knot in $S3$. Let $K$ be a knot in $S3$, and let $r$ and $r'$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; then there is no orientation preserving homeomophism between the manifolds obtained by performing Dehn surgery of type $r$ and $r'$, respectively. In particular, this result implies the Knot Complement Theorem of Gordon and Luecke.

##### The structure of groups with a quasiconvex hierarchy

We prove that hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups. Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is "virtually special", and this yields the separability of the quasiconvex subgroups of the groups we study.

##### Quasi-isometric classification of 3-manifold groups

Any finitely generated group can be endowed with a natural metric which is unique up to maps of bounded distortion (quasi-isometries). A fundamental question is to classify finitely generated groups up to quasi-isometry. Considered from this point of view, fundamental groups of 3-manifolds provide a rich source of examples. Surprisingly, a concise way to describe the quasi-isometric classification of 3-manifolds is in terms of a concept in computer science called "bisimulation." We will focus on describing this classification and a geometric interpretation of bisimulation. Finally, time permitting, we will provide applications to the study of Artin groups. (Joint work with Walter Neumann.)

##### Resonance for loop homology on spheres

A Riemannian metric on a compact manifold $M$ gives rise to a length function on the free loop space $LM$, whose critical points are the closed geodesics in the given metric on $M$. If $x$ is a homology class on $LM$, the "minimax" critical level $cr(x)$ is a critical value. Let $M$ be a sphere, and fix a metric and a coefficient field. We prove that the limit as $deg(x)$ goes to infinity of $cr(x)/deg(x)$ exists. Mark Goresky and Hans-Bert Rademacher are collaborators.