# Topology Seminar

This seminar is held on Tuesdays at 3pm in Zoom Meeting with meeting ID 979 3199 8061 and passcode 438641

### Summer, 2022

- July 7,
**John Baldwin**(Boston College)

**Time:**10am

Floer homology and non-fibered knot detection**Abstract**: A basic question one can ask about an invariant of knots is which knots (if any) the invariant detects. For example, it is a famous open question whether the Jones polynomial detects even the simplest knot, the unknot. This talk will be concerned about detection results in knot Floer homology and Khovanov homology, two knot invariants which have played central roles in low-dimensional topology over the last two decades. At present, these link homology theories are known to detect exactly six knots: the unknot, the two trefoils, the figure eight, and the two cinquefoils. One thing these knots have in common is that they are all fibered, and this fact was crucial in these detection results (at least for the nontrivial knots in the list). I’ll describe joint work with Steven Sivek in which we prove for the first time that knot Floer homology and Khovanov homology can also detect non-fibered knots, including 5_2. Furthermore, we show that HOMFLY homology can detect infinitely many such knots. These results follow from our main theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading has rank 2 and is supported in a single mod-2 homological grading.

### Spring Quarter, 2022

- March 29,
**Anubhav Mukherjee**(Georgia Tech)

Exotic surfaces and the family Bauer-Furuta invariant**Abstract**: An important principle in 4-dimesional topology, as discovered by Wall in the 1960s, states that all exotic phenomena are eliminated by sufficiently many stabilizations (i.e., taking connected sum with S2 xS2’s). Since then, it has been a fundamental problem to search for exotic phenomena that survives one stabilization. In this talk, we will establish the first pair of orientable exotic surfaces (in a puctured K3) which are not smoothly isotopic even after one stabilization. A key ingredient in our argument is a vanishing theorem for the family Bauer-Furuta invariant, proved using equivariant stable homotopy theory. This theorem applies to a large family of spin 4-manifolds and has some interesting applications in Smale’s conjecture (about exotic diffeomorphisms on S^4). In particular, it implies that the S^1-equivariant or non-equivariant family Bauer-Furuta invariant do not detect an exotic diffeomorphism on S^4 and it suggests that the Pin(2)-symmetry could be a game changer. This is a joint work with Jianfeng Lin. - April 5,
**Anna Cepek**(UO)

The geometry of Milnor’s link invariants**Abstract**: We discuss Milnor’s link invariants through a geometric lens using intersections of Seifert surfaces. Our work is thus of a similar flavor as that of Cochran from 1990, who based his work on particular choices of Seifert surfaces. But like Mellor and Melvin in 2003, who considered only the first invariant (after linking number), we allow for more arbitrary choices. We conjecture that Milnor’s invariants can be recovered geometrically using the work of Monroe and Sinha on linking of letters and Sinha and Walters on Hopf invariants. We expect our approach to recover Cochran’s work and to extend work of Polyak, Kravchenko, Goussarov, and Viro on Gauss diagrams. - April 12,
**Gary Guth**(UO)

Ribbon homology cobordisms and link Floer

homology**Abstract**: To a knot in a 3-manifold, link Floer homology associates an F[U, V]-module, and to a link cobordism between such pairs, an F[U, V]-equivariant map. The induced maps behave well under simple geometric modifications to the link cobordism, such as doing surgery or tubing on surfaces embedded in the cobordism. Using these techniques, one can prove that ribbon homology concordances induce split injections on Link Floer homology (generalizing work of Daemi—Lidman—Vela-Vick—Wong and Zemke) and give restrictions on the number critical points in ribbon homology concordances. - April 19,
**Adam Howard**(Montana State)

Framed diffeomorphisms of the torus**Abstract**: The two-dimensional torus is a parallelizable manifold and therefore can be equipped with a framing (a trivialization of its tangent bundle.) We can then consider diffeomorphisms of the torus which preserve this framing up to homotopy. The set of all diffeomorphisms which satisfy this condition forms a topological group which we call the group of framed diffeomorphisms. We will see that for a framing homotopic to a translation invariant framing (for example the standard framing) that the group of framed diffeomorphisms is homotopy equivalent to a semidirect product of a torus and the braid group on 3-strands. Identifying this group gives rise to extending natural symmetries of secondary Hochschild Homology. - April 26,
**Sarah Petersen**(Notre Dame)

Ravenel-Wilson Hopf ring methods in C_2-equivariant homotopy theory and the HF_2-homology of C_2-equivariant Eilenberg-MacLane spaces**Abstract**: This talk describes an extension of Ravenl-Wilson Hopf ring techniques to -equivariant homotopy theory. Our main application and motivation for introducing these methods is a computation of the -graded homology of -equivariant Eilenberg-MacLane spaces. The result we obtain for -equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor gives a -equivariant analogue of the classical computation due to Serre at the prime 2. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg-MacLane spaces and suggest the existence of another twisted bar spectral sequence with -page given in terms of a twisted Tor functor. - May 3,
**Inna Zakharevich**(Cornell)

Scissors congruence and regulators**Abstract**: Two polytopes are called “scissors congruent” if one can be “cut up” into finitely many pieces, and the pieces rearranged to make the other one. The question of classifying polytopes up to scissors congruence is well-understood in 1,2,3 and 4-dimensional Euclidean geometry: the important invariants are the volume and the “Dehn invariant”, a kind of three-dimensional weighted angle measure. In other geometries and dimensions more general Dehn invariants arise as well, and play an important role in relating scissors congruence to algebraic K-theory. In this talk we will introduce scissors congruence groups and the Goncharov complex (a chain complex built out of Dehn invariants), whose homology is conjecturally related to the weight filtration in algebraic K-theory. We will then describe a derived construction for the Dehn invariant and show how this construction illuminates the connection between the Goncharov complex, volume, and the Borel regulator (a volume-like morphism out of algebraic K-theory). - May 10,
**Morgan Opie**(UCLA)

Chromatic invariants of vector bundles on projective spaces**Abstract**: In this talk, I will discuss my ongoing work on complex rank 3 topological vector bundles on CP^5. I will describe a classification of such bundles using twisted, topological modular form-valued invariants, and the subtleties involved in actually computing this invariant. As time allows, I will outline future chromatic directions. - May 24,
**Junliang Shen**(Yale)

Hitchin systems, character varieties, and the P=W conjecture**Abstract**: Nonabelian Hodge theory relates topological and algebro-geometric objects associated to a Riemann surface. Specifically, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces for C: the character variety and the Hitchin moduli space.In 2010, de Cataldo, Hausel, and Migliorini proposed a conjectural relation — now called the P=W conjecture — between these two spaces. This conjecture gives a precise link between the topology of the Hitchin system and the Hodge theory of the character variety, imposing surprising constraints on each side. I will start with an introduction to this circle of ideas; then I will survey some recent progress towards understanding this conjecture. In particular, I will discuss how geometry in characteristic p play a crucial role. Based on joint work with Mark de Cataldo, Davesh Maulik, and Siqing Zhang.

- May 31,
**Laura Fredickson**(UO)

Asymptotic Geometry of the Hitchin moduli space**Abstract**: The Hitchin moduli space is a central object in mathematics. It has played a fundamental role in the study of character varieties, conformal field theory, and mirror symmetry. The Hitchin moduli space has a very rich geometric structure, and in particular is hyperkähler. I will introduce Higgs bundles and the Hitchin moduli space and we’ll consider the example of rank 1 Higgs bundles on the torus in detail. At the end, I will talk about a beautiful set of conjectures coming from physics about the asymptotic geometry of the Hitchin moduli space, and some work proving aspects of this.

### Winter Quarter, 2022

- January 18,
**Daniel Grady**(Texas Tech)

The geometric cobordism hypothesis**Abstract**: The cobordism hypothesis of Baez-Dolan, whose proof was sketched by Lurie, provides a beautiful classification of topological field theories: for every fully dualizable object in a symmetric monoidal (infinity,d)-category, there is a unique (up to a contractible choice) topological field theory whose value at the point coincides with this object. As beautiful as this classification is, it fails to include non-topological field theories. Such theories are important not just in physics, but also in pure mathematics (for example, Yang-Mills). In this talk, I will survey recent work with Dmitri Pavlov, which proves a geometric enhancement of the cobordism hypothesis. In the special case of topological structures, our theorem reduces to the first complete proof of the topological cobordism hypothesis, after the 2009 sketch of Lurie. - February 8,
**Joshua Wang**(Harvard)

Knot homologies of band sums and split link detection**Abstract**: In this talk, I’ll discuss the behavior of several knot homology theories under a general operation of adding twists to a band within a knot. There turn out to be two qualitatively different behaviors: the dimension of the homological invariant is either independent of the number of twists added to the band, or the dimension is unbounded as more twists are added. I’ll discuss applications to the cosmetic crossing conjecture and to split link detection for sl(P) link homology when P is prime. - February 22,
**Yang Hu**(UO)

Metastable complex vector bundles over complex

projective spaces**Abstract**: We study unstable topological complex vector bundles over complex projective spaces. It is a classical problem in algebraic topology to enumerate rank r bundles over (with 1 < r < n) having fixed Chern class data. A particular case is when the Chern data is trivial, which we call the vanishing Chern enumeration. In this talk, we apply a modern tool, Weiss calculus, to produce the vanishing Chern enumeration in the first two unstable cases (which belong to what we call the “metastable” range, following Mark Mahowald), namely rank (n-1) bundles over for n > 2, and rank (n-2) bundles over for n > 3. - March 1,
**Hannah Schwartz**(Princeton)

Isotopy vs. homotopy for disks with a common dual**Abstract**: Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90’s, along with homotopy theoretic invariants of Dax from the 70’s obstructing isotopy of disks. Using the Dax invariant, we will give conditions under which pairs of homotopic properly embedded disks in a smooth 4-manifold with boundary with a common dual are isotopic. - March 8,
**Robert DeYeso**(North Carolina State)

Thin knots and the Cabling Conjecture**Abstract**: The Cabling Conjecture of González-Acuña and Short holds that only cable knots admit Dehn surgery to a manifold containing an essential sphere. We approach this conjecture for thin knots using Heegaard Floer homology, primarily via immersed curves techniques inspired by Hanselman’s work on the Cosmetic Surgery Conjecture. We show that almost all thin knots satisfy the Cabling Conjecture, with possible exception coming from a (conjecturally non-existent) collection of thin, hyperbolic, L-space knots. This result also serves as a reproof that the Cabling Conjecture is satisfied by alternating knots.

### Fall Quarter, 2021

- October 26,
**Dev Sinha**(UO)

Geometric cochains and the phenotypics of homotopy**Abstract**: (joint with Greg Friedman and Anibal Medina) Interplay between discrete and continuous, combinatorial and geometric, digital and analog has always been at the heart of topology. This was expressed by Sullivan, who likened homotopy types with genetic codes, both being discrete data with continuous expressions, as he made remarkable progress in both homotopy theory and smooth topology. We are developing geometric cochains as a way, conjecturally, to provide an E-infinity algebra model – and thus homotopy model, by Mandell’s Theorem – for smooth manifolds through their differential topology. This would provide a phenotypical determination of the genetics of a manifold.Geometric cochains are a smooth version of Chow theory, developed just in the last decade by gauge theorists such as Lipyanskiy and Joyce. The theory has been “in the air” since the development of classical cohomology theories, but there are technical obstacles. With manifolds with boundaries being needed for chain complex structure, and the natural product being intersection or more generally fiber product, one is quickly led to working with manifolds with corners. Moreover, the product requires transversality and thus is partially defined, as called for if it is to be commutative while modeling some inherently E-infinity algebra. In work being written, we set the foundations of this theory. Our original motivation for studying such a theory was pedagogical – teaching basic and intermediate algebraic topology in our department – and we indicate some of those applications.

Recently posted work provides a proof of concept, where we bind the combinatorially defined cup product to the geometrically defined fiber product when both are in play – namely on a manifold with a smooth cubulation or triangulation. We do this through an almost-canonical vector field on a cubulated manifold, whose flow interpolates between the usual geometric diagonal and the Serre diagonal.

In the combinatorial setting, choices for resolving lack of commutativity at the cochain level give rise to an E-infinity structure. We think that choices for resolving lack of transversality give rise to a partially defined E-infinity structure on geometric cochains. In particular, we have an explicit conjecture for a partially defined action of the Fulton-MacPherson operad on geometric cochains. We hope to connect with experts on partially defined algebras and related matters to help resolve technicalities in this program. - November 2,
**Isaac Sundberg**(Bryn Mawr)

Geometric cochains and the phenotypics of homotopy**Abstract**: A smooth, oriented surface that is properly embedded in the 4-ball can be regarded as a cobordism between the links it bounds, namely, the empty link and its boundary in the 3-sphere. To such link cobordisms, there is an associated linear map between the Khovanov homology groups of the boundary links, and moreover, these maps are invariant, up to sign, under boundary-preserving isotopy of the surface. In this talk, we review these maps and use their invariance to understand the existence and uniqueness of slice disks and other surfaces in the 4-ball. This reflects joint work with Jonah Swann as well as Kyle Hayden. - November 9,
**Kelly Pohland**(UO)

The RO(C_3)-graded cohomology of C_3-surfaces in Z/3-coefficients**Abstract**: In this talk, we explore a recent family of computations in RO(C_3)-graded cohomology where C_3 denotes the cyclic group of order 3. In 2019, Hazel computed the RO(C_2)-graded cohomology of all C_2-surfaces in constant Z/2-coefficients based on a classification given by Dugger. We perform similar computations, instead classifying surfaces with an action of C_3 and then computing their RO(C_3)-graded cohomology in Z/3-coefficients. In this talk, we give an overview of the main result as well as demonstrate some of the techniques used through small examples. - November 16,
**Gage Martin**(Boston College)

Annular links, double branched covers, and annular Khovanov homology**Abstract**: Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this construction does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns. - November 30,
**Thomas Brazelton**(U. Pennsylvania)

Homotopy Mackey functors of equivariant algebraic K-theory**Abstract**: Given a finite group G acting on a ring R, Merling constructed an equivariant algebraic K-theory G-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction. We will examine the algebraic structure of these homotopy Mackey functors, demonstrating that restriction and transfer data arise as restriction and extension of scalars along twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980’s which constructs Mackey functors out of the algebraic K-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K-theory of endomorphism rings, the K-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings. - December 7,
**Jonathan Hanselman**(Princeton)

The surgery formula for Heegaard Floer homology via immersed curves**Abstract**: Heegaard Floer homology is a powerful invariant of closed 3-manifolds and knot Floer homology is a related invariant for knots. When a 3-manifold is obtained by Dehn surgery on a knot these invariants are related by a surgery formula. This relationship has been a valuable tool both for computing Heegaard Floer invariants of 3-manifolds and for better understanding the Dehn surgery operation, which is a fundamental method of constructing 3-manifolds. I will describe a recent reinterpretation of this surgery formula as a geometric operation on immersed curves in the torus that makes it easier to extract certain information. As an application, I will describe recent progress on the cosmetic surgery conjecture, which states that two different surgeries on a knot must produce different manifolds. In particular we can show that the conjecture holds for all but two pairs of slopes on any given knot and that it holds for all pairs of slopes unless certain rare conditions are met.

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