Topology Seminar 2018
This seminar is held on Tuesdays at 3pm in 210 Deady.
Spring Quarter, 2018
- April 24, Emmy Murphy (Northwestern)
Arboreal singularities and loose LegendriansAbstract: Arboreal singularities are a class of singular Lagrangians, originally due to Nadler, which are generic in a certain sense. On the other hand, loose Legendrians are a class of Legendrians which can approximate certain “wrinkle” singularities, and from this they satisfy an h-principle by which their isotopy classes can be completely understood. This talk will focus on the relationship between them, looking at the Legendrian which is the link of an arboreal singularity. After reviewing the basics of both of these objects, we will present the main theorem, which shows that the category of constructable sheaves detects looseness for this class of Legendrians coming from arboreal singularities.
- May 1, Cornelia Van Cott (U. San Francisco)
Continued fractions, non-orientable surfaces, and torus knotsAbstract: Continued fractions are connected to the world of 3- and 4-manifolds in some unexpected ways. In 1969, Bredon and Wood studied non-orientable surfaces which embed in 3-manifolds. In the case of lens spaces, the types of non-orientable surfaces which embed in L(p,q) depend roughly on the continued fraction expansion of p/q. Years later, Teragaito computed the minimal genus of non-orientable surfaces in S^3 bounded by torus knots and found that the answer was similarly connected to continued fraction expansions. In this talk, we will review some of these results and use the same techniques together with modern tools from knot Floer homology to study non-orientable surfaces in B^4 bounded by torus knots. This is joint work with Slaven Jabuka.
- May 8, Jennifer Hom (Georgia Tech)
Knot concordance in homology cobordismsAbstract: The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4, with the operation induced by connected sum. We consider various generalizations of the knot concordance group, and compare these to the classical case. This is joint work with Adam Levine and Tye Lidman.
- May 15, John Etnyre (Georgia Tech)
Branched covers and contact geometryAbstract: In the 1980’s Thurston proved that there is a “universal link” in the 3-sphere, this means that there is a link L such that any 3-manifold is a branched cover over the 3-sphere with branch locus L. This surprising results led to many papers further exploring universal links and knots. In this talk I will discuss branched covers and recent work with Casals that moves Thurston’s work into the category of contact manifolds. Specifically we will show that there is a transverse link T in the standard contact 3-sphere such that any contact 3-manifold can be realized as a branched cover with branch locus T.
- May 15, Victor Turchin (Kansas State)
Special Time:4pm in Deady 210
Hochschild-Pirashvili homology and representations of Out(F_n)Abstract: Higher Hochschild (co)homology is a bifunctor that assigns to a (co)commutative (co)algebra and a topological space a graded vector space. The case when the space is a circle produces the usual Hochschild (co)homology. A striking fact discovered by Pirashvili is that this homology evaluated on a sphere has a natural splitting called Hodge decomposition and the terms of the splitting up to a regrading depend only on the dimension of the sphere. We discovered that this splitting takes place on the level of complexes for any suspension, including wedge of spheres. For more general spaces, one only has a functorially defined filtration in the higher Hochschild complexes, which is the Poincare-Birkhoff-Witt filtration when one deals with a circle or a wedge of circles. But when one considers a map between suspensions or wedges of spheres, the splitting does behave as a filtration. Applying this construction to the special case of a wedge of circles, we were able to produce new representations of Out(F_n) not factoring through GL(n,Z), n>=3, and with one of them having the smallest known dimension, namely n(n^2+5)/6 versus (2^n-1)(n-1)(n-2)/2 for the previously known one. This is joint work with Thomas Willwacher.
- May 18, Victor Turchin (Kansas State)
Special Time:3pm in Deady 210
Manifold functor calculusAbstract: The functor calculus on manifolds has been developed by Goodwillie and Weiss in order to study spaces of embeddings between two manifolds. I will explain the main ideas and constructions behind this approach: polynomial functors, Taylor tower of functors, the Gromov h-principle and its Godwillie-Weiss replacement for spaces of maps avoiding singularities that depend on several points.
- May 22, Kristen Hendricks (Michigan State)
Special Time:9am in Deady 210
Connected Heegaard Floer homology and homology cobordismAbstract: We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta’s theorem. This is joint work with Jen Hom and Tye Lidman.
- May 22, Akhil Mathew (U. Chicago)
p-adic K-theory and topological cyclic homologyAbstract: Algebraic K-theory is a basic invariant of rings connected to deep phenomena in arithmetic, geometry, and topology. When one works with the p-adic K-theory of p-adic rings, the theory of trace maps and the apparatus of topological cyclic homology (TC) is often a highly effective approximation used in many computations. The theory TC, while more conceptually involved than K-theory, is often easier to work with. In this talk, I will give a gentle introduction to these theories and explain their applications to some new structural results in K-theory, which are joint with Dustin Clausen and Matthew Morrow.
- May 29, Siqi He (CalTech)
Note Joint seminar with Geometric Analysis seminar at 11am.
A Kobayashi-Hitchin correspondence for the extended Bogomonly EquationsAbstract: We will discuss Witten’s gauge theory approaches to define the Jones polynomial for a knot over general 3-manifold by counting solutions to some gauge theory equations. We will discuss a Kobayashi-Hitchin type correspondence for the dimensional reduction of these gauge equations. This talk will base on joint works with R. Mazzeo.
- June 5, Tye Lidman (NC State)
Spines in four-manifoldsAbstract: Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of four-manifolds homotopy equivalent to surfaces.
Winter Quarter, 2018
- January 16, Samantha Allen (Indiana University)
The nonorientable four-genus of knotsAbstract: The 4–genus of a knot K is the minimal genus of a surface in B^4 whose boundary is K. Similarly, we can define the nonorientable 4–genus of a knot K as the minimal “nonorientable genus” of a surface in B^4 whose boundary is K. Finding the nonorientable 4–genus of a knot can be quite intractable; existing methods exploit the relationship between nonorientable genus and normal Euler number of the nonorientable surface.
In this talk, I will give an overview of the interplay between the nonorientable genus and normal Euler number of nonorientable surfaces in B^4. I will define both of these invariants and discuss their computation for closed surfaces and then for surfaces with boundary a knot. In particular, when fixing a knot K, we can ask what pairs of nonorientable genus and normal Euler number are realizable for a surface whose boundary is K. We will see that both classical invariants and Heegaard–Floer invariants can be used towards answering this question.
- January 23, Dev Sinha (UO)
Rational mapping invariantsAbstract: We give homotopy invariants of maps from X to Y (sometimes called “homotopy periods”), which are conjectured to be a family of complete invariants when Y is a rational space and both X and Y are simply connected. We will start with a review of the Sullivan and Quillen approaches to rational homotopy theory, and then my work with Ben Walter which resolves the rational homotopy periods question when X is a sphere. We will then discuss Maurer-Cartan equations, and share progress on associating invariants to their solutions in the Lie coalgebraic setting. Concrete examples will be emphasized throughout the talk.
- January 30, Dan Dugger (UO)
Derived Morita theory in topologyAbstract: Morita theory deals with the question of when two rings have equivalent categories of modules. Rickard generalized this to answer the question of when two DGAs have equivalent derived categories of dg-modules. Schwede and Shipley then adapted these ideas to the topological context, showing that any reasonable stable model category is equivalent to the model category of modules over a ring spectrum with many objects. I will review this whole story and then talk about a possible application to equivariant topology.
- February 6, Benson Farb (University of Chicago)
How to make predictions in topology using number theoryAbstract: In this talk I will explain how Melanie Wood, Jesse Wolfson and I were led to discover some surprising (to us) coincidences in topology purely by analogy with some classical analytic number theory. These coincidences are given in terms of the “homological density” of one space in another. We have no explanation as to why these topological predictions end up being true. I will also explain why the following question is not completely crazy: “Why is the Riemann zeta function evaluated at n+1 like the 2-fold loop space of projective n-space?”
- February 13, Cynthia Lester (UO)
The homotopical canonical topologyAbstract: Topological spaces have open covers; in an analogous way, categories can have Grothendieck topologies. These topologies are often used to define sheaves on a category (in the usual manner), and thus allow us to talk about sheaf cohomology (as a derived functor). There is a special Grothendieck topology, called the canonical topology, which contains almost all of the topologies we can write down. In nice categories, the canonical topology has a concrete presentation. I will be talking about the homotopical version of this presentation with a brief discussion on Grothendieck topologies, the canonical topology, and homotopy colimits.
- February 20, Krishanu Sankar (University of British Columbia)
Steinberg summands and symmetric powers of the equivariant sphere spectrumAbstract: The mod Steenrod algebra is the (Hopf) algebra of stable operations on mod cohomology. This algebra can be computed in several possible ways: one way is to filter the Eilenberg-Maclane spectrum using the finite symmetric powers of the sphere spectrum. The cofibers of this filtration are Steinberg summands (from the representation theory of ) of the classifying spaces .
Our main result is to lift this to -equivariant stable homotopy theory, where is any finite abelian -group (the main case of interest being when is cyclic of order a power of ). We can thus compute the -equivariant Steenrod algebra by decomposing the -equivariant classifying space of – we’ll describe this computation for . When and the equivariant dual Steenrod algebra is known due to Hu-Kriz and others, but at odd primes this is new. If there is time, we will then discuss a conjectured construction of the equivariant analogues of the Milnor operations (the indecomposables in the dual Steenrod algebra).
- February 27, Kadriye Nur Saglam (UC Riverside)
New Exotic 4-manifolds via Luttinger surgery on Lefschetz fibrationsAbstract: In this talk, I will present a new construction of exotic symplectic 4-manifolds homeomorphic but not diffeomorphic to (2h+2k-1)CP^2#(6h+2k+3)(-CP)^2 via Luttinger surgery for any (h,k)≠(0,1). First, I will introduce two symplectic building blocks for our construction: 1) the family of Lefschetz fibrations on Σ_k × S^2#4(h+1)(-CP)^2, constructed by Y. Gurtas, and 2) 4-manifolds obtained from Σ_g x T^2 via Luttinger surgery. Next, I will show how to obtain the exotic copies of (2h+2k-1)CP^2 # (6h+2k+3)(-CP)^2 by gluing these building blocks. If time permits, I will also construct new symplectic 4-manifolds with the free group of finite rank and various other finitely generated groups as the fundamental group. This is a joint work with Anar Akhmedov.
- March 6, Akram Alishahi (Columbia University)
Trivial tangles, compressible surfaces, and Floer homologyAbstract: Tangles are building blocks of knots and links. In this talk, we will introduce tangles, and a notion of triviality for their components, called boundary parallelness. Then, we will sketch a way, that is checkable by computer, to detect boundary parallel components of tangles. We will also discuss the analogous question for 3-manifolds boundary: does the boundary have a (homologically essential) compressing disk? This is a joint work with Robert Lipshitz.
- March 13, Mauricio Gomez Lopez (UO)
Spaces of graphs and the stable homology of the automorphism groups of free groupsAbstract: The goal of this talk is to explain the theorem of Galatius which describes the stable homology of the automorphism groups of free groups. Namely, Galatius proves that this stable homology is isomorphic to the homology of a component of the infinite loop space corresponding to the sphere spectrum, thus showing that the automorphism groups of free groups and the symmetric groups have the same stable homology.
A novel construction which Galatius introduced in his proof are certain spaces of graphs embedded in the n-dimensional Euclidean space. These spaces are what provide the link between the sphere spectrum and the automorphism groups of free groups, and Galatius uses them to construct models for the classifying spaces of such groups. Besides outlining the proof of Galatius, I will also explain in this talk some of the main properties of these spaces of graphs.
Fall Quarter, 2017
- September 26, Nathan Perlmutter (Stanford)
Parametrized Morse Theory, Cobordism Categories, and Positive Scalar CurvatureAbstract: In this talk I will show how to use parametrized Morse theory to construct a map from the infinite loopspace of certain Thom spectrum, MTSpin(d), into the space of positive scalar curvature metrics on a closed spin manifold of dimension d > 4. My main novel construction is a cobordism category consisting of manifolds equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. My first result identifies the homotopy type of the classifying space of this category with the infinite loopspace of another Thom spectrum that is related to MTSpin(d) and the space of Morse jets on Euclidean space. This result can viewed as an analogue of the well known theorem of Galatius, Madsen, Tillmann, and Weiss, for manifolds equipped with the extra geometric structure of a choice of admissible Morse function, with critical points confined to a range of prescribed degrees.
In the second part of the talk I will show how to use this cobordism category to probe the homotopy type of the space of positive scalar curvature metrics on a closed, spin manifold M, when dim(M) > 4. This uses a parametrized version of the Gromov-Lawson construction developed by Walsh and Chernysh. Our main result detects many non-trivial homotopy groups in the space of positive scalar curvature metrics. In particular, it gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.
- October 3, John Lind (Reed)
T-duality in physics and topologyAbstract: T-duality arose as an agreement between the predictions of different versions of string theory. There is an underlying topological agreement as well, which can be expressed as an isomorphism between the twisted K-theory of certain circle bundles. I will carefully explain this through examples, and then I will discuss my work with Westerland and Sati on T-duality for more general sorts of fiber bundles. In particular, I will describe the universal T-duality theory for sphere bundles of a fixed rank and its relationship with algebraic K-theory. Time permitting, I will discuss current work on a T-duality isomorphism in chromatic homotopy theory.
- October 10, Nikolai Saveliev (U. Miami)
Instanton knot homology and equivariant gauge theoryAbstract: Singular instanton knot homology is a Floer theory defined by Kronheimer and Mrowka using via gauge theory on orbifolds; they used it to prove that Khovanov homology is an unknot detector. We show how replacing gauge theory on an orbifold with an equivariant gauge theory on its double branched cover simplifies the matters and allows for a number of explicit calculations. This is a joint work with Prayat Poudel.
- October 17, Jeremy Van Horn-Morris (U. Arkansas)
Contact and symplectic topology and the topology of 3- and 4-manifoldsAbstract: Contact and symplectic geometry give a lens to the study of manifolds that sits somewhat at the intersection or midpoint between complex and Riemannian geometry. The tools here are more flexible than in either complex or Riemannian geometry while still retaining some of the strong constraints of the first and the some of the ubiquity of the second. Particularly in 3- and 4-dimensions, contact and symplectic structures have strong connections to the underlying topology of the manifold. We’ll survey the motivating open questions in the field with a particular emphasis on 3- and 4-dimensions and along the way we’ll introduce some of the many Floer theories that have become the dominant toolkit in the modern development of the field.
- October 24, Eric Hogle (UO)
On the RO(C2)-graded cohomology of certain equivariant GrassmanniansAbstract: The Grassmannian manifold of k-planes in R^n has a group action if R^n is taken to be a real representation of the group. When the group is C2 (the cyclic group on two elemnts), the Schubert cell construction of the Grassmannian generalizes to an equivariant representation-cell structure. However, this cell structure depends an identificiation of the representation with R^n via some equivariant flag. A different choice of flag can give a very different cell structure.
I will explain a program to compute the RO(C2)-graded Bredon cohomology of these important spaces. A theorem of Kronholm dictates that these must be free modules over the cohomology of a point, but the degrees of the generators is, in general, unknown. The ambiguity introduced by the choice mentioned above turns out to be an asset for this task. I use a computation by Dan Dugger of the cohomology of an infinite equivariant Grassmannian, and prove some theorems about equivariant flag manifolds to find the cohomologies of several infinite families of finite-dimensional equivariant Grassmannians. I will present the main ideas behind how this is done.
- October 31, Clover May (UO)
A structure theorem for RO(C_2)-graded cohomology
Abstract: Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2 the cyclic group of order two. In this talk I will present a structure theorem for RO(C_2)-graded cohomology with coefficients in the constant Z/2 Mackey functor that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will sketch the proof, which depends on a Toda bracket calculation, and give some examples.
- November 7, Keegan Boyle (UO)
The virtual cosmetic surgery conjectureAbstract: The cosmetic surgery conjecture, which has been around since the early 1990s, asks when surgery along two different framing curves for the same knot produce the same 3-manifold. The virtual cosmetic surgery conjecture is a generalization of this question to coverings between surgeries. I will explain the conjecture, discuss how far you can get with elementary techniques, and present an application of hyperbolic geometry to the conjecture. Time permitting, I will also discuss the relevance of equivariant Floer homology theories.
- November 14, Beibei Liu (UC Davis)
Heegaard Floer homology of L-space links with two componentsAbstract: We compute different version of link Floer homology for any L-space link with two components and prove that they are determined by the Alexander polynomials of every sublink of the L-space link. As an application, we compute the Thurston polytope and Thurston norm of the link and give some explicit examples.
- November 21, Mauricio Gomez Lopez (UO)
The homotopy type of the PL cobordism categoryAbstract: Over the past 20 years, diffeomorphism groups and their classifying spaces have been the subject of intense research in both algebraic and geometric topology. The interest in these kinds of spaces is largely due to the Madsen-Weiss Theorem, which is a result that identifies the homology of the stable mapping class group with the homology of the infinite loop space of a certain Thom spectrum. The significance of this result, and of the previous work of Tillman that led up to it, lies in the fact that it introduced novel ways of using methods from homotopy theory in the study of automorphism groups of manifolds.
A key tool in several of the existing proofs of the Madsen-Weiss Theorem is the smooth cobordism category. By performing a systematic study of this category, Galatius and Randal-Williams produced a substantially elementary proof of the Madsen-Weiss Theorem which bypasses a lot of the heavy machinery used in the earlier proofs of this result. Moreover, Galatius and Randal-Williams later refined the methods of this proof to obtain a version of the Madsen-Weiss Theorem for higher dimensional manifolds. In this talk, besides providing a general introduction to the Madsen-Weiss Theorem, I will report on my progress in translating this body of results to the category of piecewise linear manifolds. More specifically, in this talk I will introduce a PL cobordism category, which can be viewed as the PL version of the cobordism category used in the context of diffeomorphism groups, and explain the main result that I have proven in this project. Concretely, my main result shows that the classifying space of the PL cobordism category is weak homotopy equivalent to the infinite loop space of a spectrum built out of spaces of PL manifolds, thus providing a PL analogue of a result of Galatius and Randal-Williams.
- December 5, Ben Knudson (Harvard)
Special Room: McKenzie 349
Subdivisional spaces and graph braid groupsAbstract: We develop an approach to the study of the configuration spaces of a cell complex X that is both flexible and suitable for computation. We proceed by viewing X, together with its subdivisions, as a “subdivisional space,” a kind of diagram object, which has associated to it certain diagrammatic versions of configuration spaces. These objects, which model the correct homotopy types, mix the discrete and the continuous, and they may be attacked by combining techniques drawn from discrete Morse theory and factorization homology. We apply our theory in the 1-dimensional example of a graph, obtaining an enhanced version of a family of chain models for graph braid groups originally studied by Swiatkowski. These complexes come equipped with a robust computational toolkit, which we exploit in numerous calculations, old and new. This is joint work with Byung Hee An and Gabriel Drummond-Cole.