Topology Seminar

Seminars During Academic Year 2013-14

Fall 2013

Wednesday August 21, 2013 in Rawles Hall 104

Stefan Friedl, Splittings of knot groups

We will discuss in which ways knot groups can be written as HNN extensions. This is joint work with Dan Silver and Susan Williams

Wednesday September 4, 2013 in Rawles Hall 104

Taylor McNeill, A Filtration of the Torelli Group

Wednesday September 11, 2013 in Rawles Hall 104

Patrick Haggerty, Connected sum at infinity and 4-manifolds

In this talk, we investigate connected sum at infinity, an operation introduced by Gompf on non-compact manifolds. In defining connected sum at infinity, rays are chosen in each manifold involved, just as a choice of disc is made in each manifold involved in a connected sum. In connected sum, the choice of disc does not affect the resulting manifold, but in connected sum infinity, a different choice of rays could change the resulting manifold. Such a dependence has been demonstrated for 3-manifolds. We present an infinite family of smooth 4-manifolds for which the choice of rays affects the connected sum at infinity operation, and then generalize this family to produce higher dimensional examples. These examples answer affirmatively a conjecture of Siebenmann.

Wednesday September 18, 2013 in Rawles Hall 104

Harry Baik, Laminations on the circle and hyperbolic geometry

We will see that laminations can be used to study groups acting on the circle. In particular, it is possible to characterize Fuchsian groups in terms of topological invariant laminations. It is a geometrization result in the sense that you start with a topological dynamical data and hyperbolic geometry comes out of the picture. Main motivation of this program is Thurston's universal circle theory for tautly foliated 3-manifolds.

Wednesday September 25, 2013 in Rawles Hall 104

Matt Hogancamp, A quasi-local approach to link homology

There exist many categorications of quantum link invariants, but as yet none of their “colored" versions are functorialunder 4-dimensional link cobordisms. In this talk I propose a new categorication of the colored Jones polynomial whichis likely to be functorial. The construction uses a new, quasi-idempotent chain complex which categories a multiple ofthe Jones-Wenzl projector.

Wednesday October 2, 2013 in Rawles Hall 104

Masaaki Suzuki, Meridional and non-meridional epimorphisms between knot groups

An epimorphism between knot groups is called meridional if a meridian is sent to a meridian. We determined whether there exists a meridional epimorphism between knot groups in case of the knots with up to 11 crossings. In this talk, we show some examples of non-meridional epimorphisms too.

Tuesday October 8, 2013 in Rawles Hall 104

Taehee Kim, Character varieties and twisted Alexander polynomials

For a knot, it is unknown if the twisted Alexander polynomials associated to SL(2,C)-representations of the knot group can detect fiberedness of the knot. In this talk, we will address this question from the viewpoint of the character variety of the knot group. This is joint work with Takahiro Kitayama and Takayuki Morifuji.

Wednesday October 9, 2013 in Rawles Hall 104

Vladimir Turaev, Brackets in loop algebras of manifolds

For a smooth oriented manifold M with non-empty boundary, we study the space Omega of loops in M based at a distinguished point of the boundary. Using the ideas of string topology of Chas-Sullivan, we define on the loop algebra H_*(Omega) a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For dim(M)> 2, this double bracket induces Gerstenhaber brackets in the representation algebras associated with H_*(Omega). This extends to higher dimensions the classical Poisson brackets on the moduli space of representations of the fundamental groups of surfaces.

Wednesday October 16, 2013 in Rawles Hall 104

Chuck Livingston, Projective curves and their singularities via Heegaard-Floer Invariants

Wednesday October 23, 2013 in Rawles Hall 104

M. Mandell, The homotopy theory of cyclotomic spectra

In this talk, I'll explain what a cyclotomic spectrum is and what topological cyclic homology (TC) is. I'll discuss recent work with Blumberg on developing a homotopy theory for cyclotomic spectra and (if time allows) our verification of a conjecture of Kaledin on the representability of TC.

Tuesday November 5, 2013 in Rawles Hall 104

Mark Powell, Link Slicing and Casson Towers

We give new minimal conditions for embedding topological locally flat disks, and apply these results to show some new examples of slice links.

Wednesday November 6, 2013 in Rawles Hall 104

Irina Bobkova, A resolution of the spectrum E^{hS_2^1} at the prime 2.

In this talk we will introduce chromatic approach in stable homotopy theory: the homotopy of the p-local sphere spectrum S is determined by a family of localizations L_{K(n)}S with respect to Morava K-theories K(n). Then we will discuss some computations when p, n=2. Considerable information here can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. Goerss, Henn, Mahowald and Rezk have constructed a resolution of the K(2)-local sphere at the prime 3 which allows to simplify computations of pi_*L_{K(2)}S. We will discuss a generalization of their work to the prime 2 and construct a resolution of the spectrum E^{hS_2^1}, which is closely related to the K(2)-local sphere at the prime 2.

Thursday November 7, 2013 in Rawles Hall 104

Phillip Egger, On the existence/non-existence of a v_2^{32} self-map on A_1 at the prime 2

Using the Adams spectral sequence, Adams proved that the mod 2 Moore spectrum RP^2 admits a degree 8 v_1 self-map, which is therefore v_1^4. Behrens, Hill, Hopkins and Mahowald proved that the cofiber M(1,4) of this map admits a v_2^{32} self-map. We expect that the smash product of RP^2 with CP^2 admits a v_1 self-map and that the cofiber A_1 of this map also admits a v_2^{32} self-map. Purpose of the talk will be to explain everything in an elementary fashion from computational point of view and given hands on computation of examples.

Wednesday November 20, 2013 in Rawles Hall 104

Jeff Meier, Distinguishing topologically and smoothly doubly slice knots

The purpose of this talk is to show the existence of an infinite family of slice knots that are topologically doubly slice, but not smoothly doubly slice. First, I will present some sufficient conditions for a knot to be topologically doubly slice, and show how the knots in question satisfy these criteria. Then, I will outline how the correction terms coming from Heegaard Floer homology can be used to obstruct these knots from being smoothly doubly slice. Along the way, we will encounter some very interesting open questions related to the study of doubly slice knots.

Spring 2014

Wednesday January 8, 2014 in Rawles Hall 104

Andrew Ranicki, Surgery on braids

The mathematical theory of braids was founded by Emil Artin: in fact, his 1947 Annals paper was written while he was in Bloomington. The representation theory of the braid groups is an important tool of low-dimensional topology. On the other hand, surgery theory has been an important tool of high-dimensional topology. The closure of a braid is a link. The object of this talk is to describe a surgery-theoretic approach to braids, building on the work of Gambaudo and Ghys (2005) and Bourrigan (2013) on the nonadditivity of the signatures of the closures of braids.

Wednesday February 19, 2014 in Rawles Hall 104

Heather Dye, The loop invariant

This talk is based on the paper _The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots and Links" and is joint work with Micah Chrisman (Monmouth University). We introduce a finite type invariant of virtual knots called the three loop invariant. This invariant is a Gauss diagram formula where regions of the diagram are enhanced with the relative index of the arrows. We investigate its properties with respect to symmetries and connected sums. The invariant is an analog of the Grishanov-Vassiliev finite type invariant of degree two.

Wednesday February 26, 2014 in Rawles Hall 104

Burglind Juhl Joericke, Braid invariants and elliptic fiber bundles

Take a smooth fiber bundle over an open Riemann surface whose fibers are tori (elliptic fiber bundle). We consider the question whether the bundle is isotopic to a holomorphic bundle. When the open Riemann surface is an annulus the isotopy class of the bundle is determined by an isotopy class of self-homeomorphisms of a fiber (more precisely by its conjugacy class). The answer to the question is given in terms of a single invariant, the conformal module of the conjugacy class. It turns out that this invariant is inverse proportional to the entropy of the conjugacy class. We also give a result for elliptic fiber bundles over a torus with a hole.

Wednesday March 5, 2014 in Rawles Hall 104

Jae Choon Cha, A Topological Approach to Cheeger-Gromov Universal Bounds

Wednesday March 12, 2014 in Rawles Hall 104

Rafael Zentner, Computations in spectral sequences from Khovanov homology via partial functoriality

We use a weak functoriality result to draw conclusions about the Kronheimer-Mrowka spectral sequence from reduced Khovanov homology to reduced instanton knot Floer homology and the Ozsvath-Szabo spectral sequence from reduced Khovanov homology with Z/2 coefficients to the Heegaard-Floer-hat homology of the double branched cover. In particular, we obtain results for the first mentioned spectral sequence for the (4,5) torus knot, and for both spectral sequences for three-stranded pretzel knots.(joint with Andrew Lobb)

Wednesday March 26, 2014 in Rawles Hall 104

Noah Snyder, Topological field theories, homotopy fixed points, and algebraic structures

The cobordism hypothesis gives a classification of local topological field theories with values in some (algebraic) n-category. Part of this theorem is that there's an O(n) action on the space of framed TFTs and that homotopy fixed points for actions of certain subgroups of O(n) classify TFTs with other topological structures (spin, oriented, etc.). This gives a dictionary between topological structures and algebraic structures, which I will explain via some examples in this talk. In 3-dimensions this is part of joint work with Chris Douglas and Chris Schommer-Pries, though I will spend most of this talk on the well-known 1- and 2-dimensional situations.

Thursday April 3, 2014 in Rawles Hall 104

Matt Hedden, Floer homology and Fractional Dehn twists

There is a rational valued invariant of an automorphism of a surface with a single boundary component called the fractional Dehn twist coefficient. Roughly, it measures the twisting of the automorphism around the boundary. The fractional Dehn twist coefficient is related to the theory of taut foliations, essential laminations, and contact structures on 3-manifolds obtained by performing Dehn surgery on fibered knots. These connections arise by associating to a fibered knot in a 3-manifold the fractional Dehn twist coefficient of the monodromy of the fibration on its complement. I'll describe how the Heegaard Floer homology of a 3-manifold bounds the fractional Dehn twist coefficient of any of its fibered knots, and some consequences this has for contact structures on 3-manifolds. This is joint work with Tom Mark.

Wednesday April 9, 2014 in Rawles Hall 104

Paul Melvin, Stable isotopy in 4-manifolds

It is a well known principle in 4-dimensional topology that homeomorphic smooth simply-connected 4-manifolds become diffeomorphic after 'stabilizing' by connected summing with S^2 x S^2 sufficiently many times. In fact many explicit examples are known for which exactly one stabilization is required, and surprisingly, no examples have been shown to require more than one. Perhaps this exoticity dissolves quickly under stabilization. An analogous principle holds for 2-spheres embedded in a simply-connected 4-manifold, namely, any two that are topologically isotopic become smoothly isotopic after stabilizing the manifold sufficiently many times. Until now, however, bounds on the number of stabilizations needed have not been found for any examples. In this talk, we will describe families of topologically isotopic knots that are not smoothly isotopic, but become so after one stabilization. This is joint work with Auckly, Kim and Ruberman.

Wednesday April 16, 2014 in Rawles Hall 104

Will Kazez, Approximating C^0 foliations

Taut foliations and tight contact structures are important topological structures on 3-manifolds that are at opposite ends of the spectrum in the study of 2-plane distributions. One is integrable everywhere, the other is integrable nowhere. Eliashberg and Thurston have shown that they are closely related; in fact, every sufficiently smooth taut foliation can be perturbed to a tight contact structure. In joint work with Rachel Roberts, we show that the smoothness assumptions on the foliation can be removed. This allows the approximation theorem to be applied to a wide range of recent constructions of (non-smooth) foliations.

Tuesday April 22, 2014 in 104 Rawles Hall

Qayum Khan, Classification of free actions of Cp on S1xSn

Let p be an odd prime, and let n be a positive integer. We classify the set of equivariant homeomorphism classes of free C_p-actions on the product S^1 x S^n of spheres, up to indeterminacy bounded in p. The description is expressed in terms of number theory. The techniques are various applications of surgery theory and homotopy theory, and we perform a careful study of h-cobordisms. The p=2 case was completed by B Jahren and S Kwasik (2011). The new issues for the odd p case are the presence of nontrivial ideal class groups and a group of equivariant self-equivalences with quadratic growth in p. The latter is handled by the composition formula for structure groups of A Ranicki (2009).

Wednesday April 23, 2014 in Rawles Hall 104

Jonathan Hillman, Seifert fibered 2-knots

An n-knot manifold is the closed manifold obtained by elementary surgery on an n-knot K:S^n to S^{n+2}. (If n=1, we assume that the surgery is longitudinal, i.e., 0-framed.) Motivated by earlier work on geometries on 2-knot manifolds, we consider when such manifolds are Seifert fibred over 2-dimensional orbifolds. Consideration of the possible base orbifolds leads to a class of 2-knots with torsion-free solvable groups, hitherto overlooked, which completes the list of such groups. This is joint work with Jim Howie of Heriot-Watt University in Edinburgh.

Wednesday April 30, 2014 in Rawles Hall 104

Corrin Clarkson, Three-manifold mutations detected by Heegaard Floer homology

Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of the Heegaard Floer homology of M differs from that of its mutant.