Brandeis Topology Seminar, Spring 2022

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Ruth Charney (charney@brandeis.edu), Kiyoshi Igusa (igusa@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

January 25: No Seminar

February 1: No Seminar

February 8: Rachael Boyd (University of Cambridge) Via Zoom

Title: Homological stability for Temperley-Lieb algebras

Abstract: Many sequences of groups and spaces satisfy a phenomenon called 'homological stability'. I will present joint work with Hepworth, in which we abstract this notion to sequences of algebras, and prove homological stability for the sequence of Temperley-Lieb algebras. The proof uses a new technique of 'inductive resolutions', to overcome the lack of flatness of the Temperley-Lieb algebras. I will also introduce the 'complex of planar injective words' which plays a key role in our work. Time permitting, I will explore some connections to representation theory and combinatorics that arose from our work.

February 15: Marissa Miller (UIUC) In person

Title: Hierarchical hyperbolicity and stability in handlebody groups

Abstract: In this talk, we explore the geometry of the handlebody group, i.e. the mapping class group of a handlebody. These groups can be viewed as subgroups of surface mapping class groups and on the surface seem similar, but based on the current state of research, the geometry of handlebody groups appears to be very different than the geometry of surface mapping class groups. In this talk we will explore two different geometric notions: hierarchical hyperbolicity (of which surface mapping class groups are the prototype), and stable subgroups, which have a nice characterization in the surface mapping class groups in terms of the orbit map to the curve graph. I will discuss how the genus two handlebody group is also hierarchically hyperbolic and has an analogous stable subgroup characterization, and I will also discuss what goes wrong in the higher genus cases that prevents hierarchical hyperbolicity and the existence of an analogous stable subgroup characterization.

February 22: No seminar, February break

March 1: Bernard Badzioch (University of Buffalo)

Title: Categorical algebra and mapping spaces

Abstract: Many classical results in homotopy theory show that iterated loop spaces, i.e. pointed mapping spaces

from a sphere, can be identified with spaces equipped with a certain algebraic structure described by means

of an operad, a prop, an algebraic theory etc. A natural questions is whether analogous algebraic description

can be used to characterize mapping spaces with the domain given by a space different than a sphere.

The talk will describe some results in this area.

March 10 (Thursday), 3:30-4:30: Jingyin Huang (Ohio State) Via Zoom

Title: The Helly geometry of some Garside and Artin groups

Abstract: Garside groups and Artin groups are two generalizations of braid groups. We show that weak Garside groups of finite type and FC-type Artin groups acts geometrically metric spaces which are non-positively in an appropriate sense, i.e. they act geometrically on Helly graphs, as well as metric spaces with convex geodesic bicombings. We will also discuss several algorithmic and , geometric and topological consequences of the existence of such an action. This is joint work with D. Osajda.

March 15: Lev Tostopyat-Nelip (Michigan State)

Title: Floer homology and quasipositive surfaces

Abstract: Ozsvath and Szabo have shown that knot Floer homology detects the genus of a knot - the largest Alexander grading of a non-trivial homology class is equal to the genus.

We give a new contact geometric interpretation of this fact by realizing such a class via the transverse knot invariant introduced by Lisca, Ozsvath, Stipsicz and Szabo. Our approach relies on the "convex decomposition theory" of Honda, Kazez and Matic - a contact geometric interpretation of Gabai's sutured hierarchies.

We use this new interpretation to study the "next-to-top" summand of knot Floer homology, and to show that Heegaard Floer homology detects quasi-positive Seifert surfaces. Some of this talk represents joint work with Matthew Hedden.

March 22: Dave Auckly (Kansas State) (tentative)

Title: Branched covers in low dimensions.

Abstract: This talk will begin with several basic examples of branched covers. It will then present several results about the existence and non-existence of branched covers in low dimensional settings.

March 29: Lei Chen (University of Maryland)

Title: Actions of Homeo and Diffeo groups on manifolds

Abstract: In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

April 5: Maria Cumplido (University of Seville)

Title: Conjugacy stability in Artin groups

Abstract: Artin (or Artin-Tits) groups are generalizations of braid groups that are defined using a finite set of generators $S$ and relations $abab\cdots=baba\cdots$, where both words of the equality have the same length. Although this definition is quite simple, there are very few results known for Artin groups in general. Classic problems as the word problem or the conjugacy problem are still open. In this talk, we study a problem concerning a family of subgroups of Artin groups: parabolic subgroups. These subgroups have proven to be useful when studying Artin groups (for example, they are used to build interesting simplicial complexes), but again, we do not know much about them in general. Our problem will be the following: Given two conjugate elements of a parabolic subgroup $P$ of an Artin group $A$, are they conjugate via an element of $P$? This is called the conjugacy stability problem. In 2014, González-Meneses proved that this is always true for braids, that is, geometric embeddings of braids do not merge conjugacy classes. In an article with Calvez and Cisneros de la Cruz, we gave a classification for spherical Artin groups and proved that the answer to the question is not always affirmative. In this talk, we will explain how to give an algorithm to solve this problem for every Artin group satisfying three properties that are conjectured to be always true.

April 12: Aleksandra Kjuchukova (Notre Dame)

Title:

Abstract:

April 19: Spring break--no seminar

May 10: Biji Wong (Duke)

NOTE: Special time (3:30-4:30) and room (317 Goldsmith)

Title: Double branched covers of links, d-invariants, and signatures

Abstract: From Heegaard-Floer theory for 3-manifolds, one can extract certain numerical invariants called d-invariants. They give information about surgeries, when 3-manifolds bound 4-manifolds, and knot concordance. In this talk, we will relate the d-invariants of double branched covers of (multi-component) links in the 3-sphere to the signatures of the branched links. This generalizes work of Manolescu-Owens, Lisca-Owens and others, and is work in progress with Marco Marengon.