# Seminars & Events for Topology Seminar

##### Bordered Heegaard Floer homology with torus boundary via immersed curves

I will describe a geometric interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In particular these invariants, originally defined as homotopy classes of modules over a particular algebra, can be described as collections of decorated immersed curves in the boundary of M. Pairing two bordered Floer invariants corresponds to taking the Floer homology of immersed curves; in most cases this simply counts the minimal intersection number. This framework leads to elegant proofs of several interesting results about closed 3-manifolds. As one example, I will prove a lower bound for the complexity of Heegaard Floer homology of a manifold containing an incompressible torus, reproving and strengthening a recent result of Eftekhary.

##### Complex curves through a contact lens

Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools.

We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

##### Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF

A group G is called LERF if the property that an element not lying in a finitely generated subgroup is visible via a finite quotient of G. LERFness of groups is closed related with low-dimensional topology: whether an immersed \pi_1-injective object can be lifted to embedding in some finite cover. We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.

##### Palindromes in Teichmuller Theory

It is well known that every primitive word in a rank two free group is conjugate to either a palindrome or the product of two palindromes. We present an enumeration scheme that gives each primitive as the unique palindrome in its conjugacy class or as a product of two unique palindromes that have already appeared in the enumeration scheme. We use this result to obtain some necessary and sucient geometric discreteness conditions in PSL(2;C) (equivalently Isom+(H3)). This is joint work with L. Keen.

##### Mayer-Vietoris sequence for relative symplectic cohomology

I will first recall the definition of an invariant that assigns to any compact subset K of a closed symplectic manifold M a module SH_M(K) over the Novikov ring. I will go over the case of M=two sphere to illustrate various points about the invariant. Finally I will state the Mayer-Vietoris property and explain under what conditions it holds.

##### The bipolar filtration of topologically slice knots

The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. We show that the graded quotient of the bipolar filtration has infinite rank at each stage greater than one.

To detect nontrivial elements in the quotient, the proof uses higher order amenable Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term $d$-invariants simultaneously.

This is joint work with Jae Choon Cha.

##### Peculiar modules for 4-ended tangles

A peculiar module is a certain algebraic invariant of 4-ended tangles that I developed in my PhD thesis as a tool for studying the local behaviour of Heegaard Floer homology for knots and links. I will briefly explain its construction and describe its classification in terms of immersed curves on a 4-punctured sphere as well as a glueing formula. Finally, I will discuss some applications, such as rational tangle detection, skein relations and mutation symmetries.