2018 Fall Meeting

University of Oregon

Eugene, OR

Saturday and Sunday, November 10 and 11, 2018

### Confirmed speakers:

Dan Christensen (Western Ontario)

Higher Toda brackets and the Adams spectral sequence

**Abstract**: I will review the construction of the Adams spectral sequence in a triangulated category equipped with a projective or injective class. Then, following Cohen and Shipley, I’ll explain how to define higher Toda brackets in a triangulated category. The main result is a theorem which gives a relationship between these Toda brackets and the differentials in the Adams spectral sequence. I will also mention a result due to Heller and Muro that shows that the 3-fold Toda bracket determine the triangulation and hence the higher Toda brackets. This is joint work with Martin Frankland.

Jeanne Duflot (Colorado State)

A Degree Formula for Equivariant Cohomology

**Abstract**: I will talk about a generalization of a result of Lynn on the “degree” of an equivariant cohomology ring

. The degree of a graded module is a certain coefficient of its PoincarĂ© series. The main theorem is an additivity formula for degree:

This work is joint with Mark Blumstein.

Mike Hill (UCLA)

Norms, transfers, and the equivariant Steenrod algebras

**Abstract**: For groups larger than

, little is known about the equivariant Steenrod algebra for constant Mackey functor coefficients. Using the norm and the norms of the Fujii–Landweber spectrum of Real bordism, we can describe a method for how one can describe what happens for cyclic 2-groups. Along the way, I’ll discuss the algebraic consequences of the norm and several twisted variants of associative algebras in the equivariant context.

Lisa Piccirillo (Texas)

The Conway knot is not slice

**Abstract**: Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. I’ll give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Jenny Wilson (Michigan)

Stability in the homology of configuration spaces

**Abstract**: This talk will illustrate some patterns in the homology of the configuration space Fk(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena — relationships between unstable homology classes in different degrees — established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius–Kupers–Randal-Williams.

Shida Wang (Oregon)

The smooth concordance group and knot Floer invariants

**Abstract**: This talk will start from an expositive introduction to the smooth knot concordance group. Then we will review knot Heegaard Floer theory and introduce a few invariants arising from the knot Floer complex. In particular, we will focus on the epsilon invariant defined by Hom and the Upsilon invariant defined by Ozsvath, Stipsicz and Szabo. Finally, we will survey a few recent results in concordance theory as applications and see L-space knots provide many examples.

### Schedule:

Saturday, November 10

- 9-10 Breakfast and registration
- 10-11 Christensen
- 11:30-12:30 Wilson
- 12:30-2 Lunch
- 2-3 Wang
- 3-3:30 Tea break
- 3:30-4:30 Duflot

Sunday, November 11

- 8:30-9 Breakfast and registration
- 9-10 Piccirillo
- 10-10:30 Break
- 10:30-11:30 Hill

Talks will be in Fenton Hall, Room 110, on the University of Oregon Campus. Fenton Hall houses the mathematics department. Other gatherings will be in the Fenton lounge, on the second floor.

The Cascade Topology Seminar is supported by the National Science Foundation along with host institutions. There is no registration fee. We encourage anyone interested in topology to attend, especially those from historically underrepresented groups. Funds are available to support the attendance of graduate students and other early career mathematicians. If you are interested in support or have any other questions, please e-mail Dev Sinha at dps@uoregon.edu

There are many hotels close to campus, and Hyatt and EVEN hotels have shuttles to the University. Eugene is served by an airport (EUG) with direct flights to major western cities. You will need to arrange for cabs to and from the airport as there is no public transportation. Portland and its airport (PDX) are two hours away by car, and there are shuttles which go from campus to PDX (in more like three hours total).