Schedule for: 17w5011 - Thirty Years of Floer Theory for 3-Manifolds

Following up on a recent talk I gave at Skye, I'll discuss a topological invariant associated to a left order on the fundamental group of a prime, closed, oriented 3-manifold. I'll also describe some instances in which a left order with non-vanishing invariant can be deformed to an order giving rise to a taut foliation, and I'll characterize which foliations can be achieved by such constructions.

Let $\{K_n\}$ be the family of knots obtained by twisting a knot K along an unknot c. When the winding number of K about c is non-zero, we show the limit of $g(K_n)/g_4(K_n)$ is 1 if and only if the winding and wrapping numbers of K about c are equal. When equal, this leads to a description of minimal genus Seifert surfaces of $K_n$ for $|n|\gg0$ and eventually to a characterization of when c is a braid axis for K. We then use this characterization to show that satellite L-space knots are braided satellites*. This is joint work with Kimihiko Motegi that builds upon joint work with Scott Taylor. (* Modulo a conjecture whose solution by Hanselman-Rasmussen-Watson has been announced.)

We will discuss a TQFT for the full link Floer complex, involving decorated link cobordisms. It is inspired by Juhasz's TQFT for sutured Floer homology. We will discuss how the TQFT recovers standard bounds on concordance invariants like Ozsvath and Szabo's tau invariant and Rasmussen's local h invariants (which are normally proven using surgery theory) and also gives a new bound on Upsilon. We will also see how well known maps in the link Floer complex can be encoded into decorations on surfaces, and as an example we will see how Sarkar's formula for a mapping class group action on link Floer homology is recovered by some simple pictorial relations. Time permitting, we will also discuss how these pictorial relations give a connected sum formula for Hendricks and Manolescu's involutive invariants for knot Floer homology.

Using Alexander modules, one can define a polynomial invariant for a certain class of graphs with a balanced coloring. We will give different interpretations of this polynomial by Kauffman state formula and MOY relations. Moreover, there is a Heegaard Floer homology of graphs whose Euler characteristic is the Alexander polynomial. This is joint work with Yuanyuan Bao.

In this talk we survey the known connections and evidence supporting the conjectured equivalence of the following three properties of a closed, connected, orientable, irreducible 3-manifold W: (i) W admits a co-oriented taut foliation; (ii) W has a left-orderable fundamental group; (iii) W is a Heegaard-Floer L-space. In particular, we discuss the relativisation of the conjectures which led to the confirmation of the conjecture for graph manifolds, and the subsequent open problems suggested by the work of Jonathan Hanselman, Jake Rasmussen, Sarah Rasmussen and Liam Watson.

I will survey applications of Floer homology to Dehn surgery over the past thirty years.

A conjecture of Akbulut and Kirby from 1978 states that the concordance class of a knot is determined by its 0-surgery. In 2015, Yasui disproved this conjecture by providing pairs of knots which have the same 0-surgeries yet which can be distinguished in (smooth) concordance by an invariant associated to the four-dimensional traces of such a surgery. In this talk, I will discuss joint work with Lisa Piccirillo in which we construct many pairs of knots which have diffeomorphic 0-surgery traces yet some of which can be distinguished in smooth concordance by the Heegaard Floer d-invariants of their double branched covers. If time permits, I will also discuss the applicability of this work to the existence of interesting invertible satellite maps on the smooth concordance group.

We give constraints on when the $n$-fold cyclic branched cover $\Sigma_n(L)$ of a strongly quasipositive link $L$ can be an L-space. In particular we show that if $K$ is a non-trivial L-space knot and $\Sigma_n(K)$ is an L-space then (1) $n \le 5$; (2) if $n$ = 4 or 5 then $K$ is the torus knot $T(2,3)$; (3) if $n$ = 3 then $K$ is either $T(2,3)$ or $T(2,5)$, or $K$ is hyperbolic and has the same Alexander polynomial as $T(2,5)$; (4) if $n$ = 2 then $\Delta_{K}(t)$ is a non-trivial product of cyclotomic polynomials. (This is joint work with Michel Boileau and Steve Boyer.)

I will describe a construction of (codimension one) co-oriented taut foliations (CTFs) of 3-manifolds. It follows from this construction that if K is a composite, alternating, or Montesinos knot, then the L-space conjecture of Ozsvath and Szabo holds for any 3-manifold obtained by Dehn surgery along K. I will focus on the alternating knot case. This work is joint with Charles Delman.

In his seminal paper on his "Jones polynomial homology," Khovanov identifies Floer homology as one of the underlying motivations for his work. In the 20 years since then, the connections between Khovanov homology (broadly interpreted) and Floer homology have been a fruitful topic for research, but there is still much that remains mysterious. I'll describe some of our successes at understanding the relations between the two theories over the last 20 years, and speculate on where things may go in the next 20.

Involutive Heegaard Floer homology is a variant on the 3-manifold invariant Heegaard Floer homology which incorporates the data of the conjugation symmetry on the Heegaard Floer complexes, and is in principle meant to correspond to $\mathbb{Z}_4$-Seiberg Witten Floer homology. It can be used to obtain two new invariants of homology cobordism and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We introduce involutive Heegaard Floer homology and its associated invariants and use it to give a new criterion for an element in the integer homology cobordism group to have infinite order, similar but not identical to a recent criterion given by Lin-Ruberman-Saviliev. Much of this talk is joint work with C. Manolescu; other parts are variously also joint with I. Zemke or with J. Hom and T. Lidman.

I will begin with a survey of concordance invariants constructed from Heegaard Floer homology. I then will describe a sequence of concordance invariants $\nu_n$ constructed from truncated Heegaard Floer homology that generalize the Ozsvath-Szabo $\nu$ invariant and the Hom-Wu $\nu^+$ invariant.

I will describe a characterization of which simple knots in lens spaces fiber. This is joint work with John Luecke.

We will describe a new interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In our setting, these invariants, originally defined as homotopy classes of (differential) modules over a particular algebra, take the form of homotopy classes of immersed curves in the boundary of M decorated with local systems. Moreover, pairing two bordered Floer invariants corresponds to taking the Floer homology of two sets of decorated immersed curves. In most cases this simply counts the minimal intersection number, which leads to a number of applications. This is joint work with Liam Watson and Jake Rasmussen.

This talk will describe some of the key components of theorem stated in the previous talk, namely, the passage from differential modules over the torus algebra to immersed curves. We give a description of type D structures in terms of train tracks in the (marked) torus, and develop a yoga for simplifying these train tracks to immersed curves with local systems. This is joint work with Jonathan Hanselman and Jake Rasmussen.

I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. Our invariant is a model for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid.

The Floer theory of a cotangent fiber in a symplectic cotangent bundle T*M can be understood via the topology of the manifold M. More generally, a Weinstein manifold has a core isotropic skeleton and we can try to understand the symplectic topology of the Weinstein manifold in terms of the topology of the skeleton. Unfortunately, generically the skeleton has singularities which make its topology difficult to understand and which lead to loss of information. We will discuss a nice minimal set of singularities which we can understand combinatorially, and try to show that all Weinstein manifolds can be deformed to have a skeleton with only these nice singularities. These singularities coincide with Nadler’s “arboreal singularities” where Floer theoretic calculations can locally be done combinatorially.

The knot concordance group has been the subject of much study since its introduction by Ralph Fox and John Milnor in 1966. One might hope to generalize the notion of a concordance group to links; however, the immediate generalization to the set of links up to concordance does not form a group since connected sum of links is not well-defined. In 1988, Jean Yves Le Dimet defined the string link concordance group, where a link is based by a disk and represented by embedded arcs in D^2 × I. In 2012, Andrew Donald and Brendan Owens defined groups of links up to a notion of concordance based on Euler characteristic. However, both cases expand the set of links modulo concordance to larger sets and each link has many representatives in these larger groups. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the the “knotification” construction of Peter Ozsv´ath and Zoltan Szab´o, giving a definition of a link concordance group where each link has a unique group representative. I will also present invariants for studying this group using both group theory and Heegaard-Floer Homology.

Given a Lie group G acting on a symplectic manifold preserving a pair of Lagrangians setwise, I will describe a construction of G-equivariant Lagrangian Floer homology. This does not require G-equivariant transversality, which allows the construction to be flexible. Time permitting, I will talk about applying this for the O(2)-action on Seidel-Smith's symplectic Khovanov homology. This is joint with Kristen Hendricks and Robert Lipshitz.