# Topology Seminar

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

### 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.

### 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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