# Seminars & Events for Topology Seminar

##### Homology of the curve complex and the Steinberg module of the mapping class group

The homology of the curve complex is of fundamental importance for the homology of the mapping class group. It was previously known to be an infinitely generated free abelian group, but to date, its structure as a mapping class group module has gone unexplored. I will give a resolution for the homology of the curve complex as a mapping class group module. From the presentation coming from the last two terms of this resolution I will show that this module is cyclic and give an explicit single generator. As a corollary, this generator is a homologically nontrivial sphere in the curve complex.

##### Holomorphic traingle maps in sutured Floer homology

Honda, Kazez and Matic defined maps on sutured Floer homology induced by a contact structure. I'll explain how to compute these maps using holomorphic triangle counts and give some applications to computing sutured Floer homologies and sutured contact invariants.

##### On local combinatorial formulae for Pontryagin classes

The talk will be devoted to the problem of combinatorial computation of the rational Pontryagin classes of a triangulated manifold. This problem goes back to the famous work by A. M. Gabrielov, I. M. Gelfand, and M. V. Losik (1975). Since then several different approaches to combinatorial computation of the Pontryagin classes have been suggested. However, these approaches require a combinatorial manifold to be endowed with some additional structure such as smoothing or certain its discrete analogue. We suggest a new approach based on the concept of a *universal local formula*. This approach allows us to construct an explicit combinatorial formula for the first Pontryagin class that can be applied to any combinatorial manifold without any additional structure.

##### Fox re-embedding and Bing submanifolds

Let $M$ be an orientable closed connected 3-manifold, and $Y$ be a connected compact 3-manifold. We show that the following two conditions are equivalent: (i) $Y$ can be embedded in $M$ so that the closure of the complement of the image of $Y$ is a union of handlebodies; and (ii) $Y$ can be embedded in $M$ so that every embedded closed loop in $M$ can be isotoped to lie within the image of $Y$. Our result can be regarded as a common generalization of Fox's *reimbedding theorem* (1948) and Bing's characterization of *3-sphere* (1958), as well as more recent results of Hass and Thompson (1989) and Kobayashi and Nishi (1994).

##### Transverse homology

Knot contact homology is a combinatorial Floer-theoretic knot invariant derived from Symplectic Field Theory. I'll discuss the geometry behind this invariant and a new filtered version, transverse homology, which turns out to be a fairly effective invariant of transverse knots

##### Somewhat simple curves on surfaces, and the mysteries of covering spaces

I will count some curves on 2-dimensional manifolds, and will discuss some related issues in geometric (and otherwise) group theory.

##### Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities.

Consider a planar, bounded, $m$-connected region $\Omega$, and let $\partial\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\partial\Omega$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.

The subject has an interesting history that started with Dehn (1903). References may be found here (#18 & #19).

##### Gromov's knot distortion

Gromov defined the distortion of an embedding of $S^1$ into $R^3$ and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of $S^1$ into $R^3$ with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will show how to give a nontrivial lower bound on the distortion of torus knots. I will also mention some natural conjectures about the distortion, for example that the distortion of the $(2,p)$-torus knots is unbounded.

##### Holomorphic Pairs of Pants in Mapping Tori

We consider invariants of mapping tori of symplectomorphisms of a symplectic surface S, such as symplectic field theory, contact homology, and periodic Floer homology, for the standard stable Hamiltonian structure on the mapping torus. These invariants involve counts of holomorphic curves in R times the mapping torus. We obtain a number theoretic description of all rigid holomorphic curves in the case S = T2, and obtain various pair-of-pants invariants for symplectomorphisms on higher genus surfaces. Our method involves reinterpreting counts of holomorphic pairs of pants in R times the mapping torus as counts of index -1 triangles between Lagrangians in S x S for certain 1-parameter families of almost-complex structures.

##### Filtering smooth concordance classes of topologically slice knots

Cochran, Orr, and Teichner introduced the n-solvable filtration of the knot concordance group, which has given a framework for recent advances in the study of knot concordance. However, it fails to detect anything about topologically slice knots, denoted T. We define a new filtration of the knot concordance group, relate it to known concordance invariants, and use Heegaard Floer homology to prove that it induces a non-trivial filtration on T. One application of this filtration is to say more about the fractal nature of the knot concordance group, i.e. the complexity of the Cochran-Orr-Teichner filtration embeds into T. This is joint work with Tim Cochran and Shelly Harvey.

##### Bordered Floer homology and the contact category

Bordered Heegaard Floer homology is a verison of Heegaard Floer homology for 3-manifolds with boundary, developed by Lipshitz, Ozsvath, and Thurston. A key component of the theory is a DG-algebra associated to a parametrized surface $F$. I will discuss how the homology of this algebra can be naturally identified with a full subcategory of the category of contact structures on $Fx[0,1]$, with convex boundary conditions.

##### Right-angledness, flag complexes, asphericity

I will discuss three related constructions of spaces and manifolds and then give necessary and sufficient conditions for the resulting spaces to be aspherical. The first construction is the polyhedral product functor. The second construction involves applying the reflection group trick to a "corner of spaces". The third construction involves pulling back a corner of spaces via a coloring of a simplicial complex. The two main sources of examples of corners which yield aspherical results are: 1) products of aspherical manifolds with (aspherical) boundary and 2) the Borel-Serre bordification of torsion-free arithmetic groups which are nonuniform lattices.

##### Nondistortedsubgroups of $Out(F_n)$, via Lipschitz retraction in spaces of trees (joint work with M. Handel)

We prove that various subgroups of $Out(F_n)$ — the outer automorphism group of a free group of rank $n$ — such as the stabilizer of the conjugacy class of a rank $n-1$ free factor, are undistorted in $Out(F_n)$. The method of proof is to show that these subgroups are Lipschitz retracts of the ambient group, in fact we construct these retractions in appropriate spaces of trees on which $F_n$ acts.

##### Unlink detection and the Khovanov module

Kronheimer and Mrowka recently showed that Khovanov homology detects the unknot. Their proof does not obviously extend to show that Khovanov homology detects unlinks of more than one component, and one could reasonably question whether it actually does (the Jones polynomial, for instance, does not detect unlinks with multiple components). In this talk, I'll discuss how to use a spectral sequence of Ozsvath and Szabo in conjunction with Kronheimer and Mrowka's result to settle the question (in the affirmative). This is joint work with Yi Ni.

##### Pseudo-Anosov flows in Seifert fibered and solvable 3-manifolds

We discuss the following rigidity results: 1) A pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically conjugate to a geodesic flow; 2) A pseudo-Anosov flow in a solv manifold is topologically conjugate to a suspension Anosov flow. The proofs use the structure of the fundamental groups in these manifolds and the topological theory of pseudo-Anosov flows. In particular the proofs use in essential ways the Z or Z+Z normal subgroups of the fundamental group. These normal subgroups interact with the orbit space of the flow or the leaf spaces of the stable/unstable foliations, producing invariant axes and chains of lozenges, which help force the rigidity. This is joint work with Thierry Barbot.

##### Pseudo-Anosov maps with small dilatation

Fix an orientable surface $S$. It is known that the set of dilatations of all pseudo-Anosov maps acting on $S$ is a family of real numbers that is bounded below by 1, and has a minimum value $\lambda_{min,S}>1$ which is realized geometrically. We will discuss recent work on the problem of determining $\lambda_{min,S}$ and show how a little-known theorem, the 'Coefficient Theorem for Digraphs,' can be used to gain insight into this set. The study of small dilatation pA maps appears to be related to the study of small volume fibered hyperbolic 3-manifolds, and an example from 3-manifolds has played a role in understanding the dilatation problem.

##### Geometric structures on moment-angle manifolds

Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra. Moment-angle complexes corresponding to simplicial subdivision of spheres are topological manifolds, and those corresponding to simplicial polytopes admit smooth realisations as intersection of real quadrics in $C^m$. After an introductory part describing the general properties of moment-angle complexes we shall concentrate on the complex-analytic and Lagrangian aspects of the theory.

##### The Rank versus Genus Conjecture

We construct a counterexample to the Rank versus Genus Conjecture (also known as the Rank Conjecture), i.e., a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus.

##### A combinatorial spanning tree model for knot Floer homology

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

##### Computational aspects of bordered Floer homology

I will give a brief outline of bordered Floer homology, and explain how it can be used to describe aspects of Heegaard Floer homology. I will pay special emphasis on the branched double-covers of links in $S3$. This is joint work with Robert Lipshitz and Dylan Thurston.