UCLA Topology Workshop 2018

• Date: January 3-5, 2018
• Location: IPAM, UCLA
• Organisers: Ko Honda, Ciprian Manolescu, Yi Ni, and Sucharit Sarkar
• Primarily funded by: NSF DMS-1643401
• With support from: UCLA Mathematics Department and IPAM

The workshop focused on recent developments in low-dimensional topology, with a particular emphasis on modern Floer homological invariants such as Heegaard Floer homology. The workshop was open to all; the following photo captures most of the participants.

Schedule

All talks were held at the Institute for Pure and Applied Mathematics (map of its location). Select a talk title to show or hide its abstract.

Wednesday, January 3, 2018

• 07:30-08:30: Breakfast
• 08:30-09:30: András Stipsicz (Alfréd Rényi Institute)
  Upsilon type invariants of knots

  We review the definition of the Upsilon invariant of a knot, show some variants of the definition and some applications.

• 09:45-10:45: Jennifer Hom (Georgia Tech)
  Heegaard Floer homology and homology cobordism, part one

  We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we associate a module ${\mathrm{HF}}_{\mathrm{conn}}(Y)$, called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of ${\mathrm{HF}}_{\mathrm{red}}(Y)$. The definition of this invariant relies on involutive Heegaard Floer homology. This is joint work with Kristen Hendricks and Tye Lidman.

• 11:00-12:00: Kristen Hendricks (Michigan State U)
  Heegaard Floer homology and homology cobordism, part two

  Continuing from Jen Hom's talk, we compute some basic examples of connected Heegaard Floer homology, and use it to derive various applications to homology cobordism. We show, for example, that an integer homology sphere whose connected homology has dimension one always has infinite order in the homology cobordism group; we also construct a new filtration of the homology cobordism group and use it to give a reproof of Furuta's theorem. We also study the limiting behavior of numerical invariants from involutive Heegaard Floer homology under connected sum. This is joint work with Jen Hom and Tye Lidman.

• 12:00-13:45: Lunch break
• 13:45-14:45: John Baldwin (Boston College)
  Khovanov homology detects the trefoil

  I'll discuss joint work with Steven Sivek in which we prove that Khovanov homology detects the trefoil. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a new bypass exact triangle in sutured instanton homology; and Kronheimer and Mrowka's spectral sequence relating Khovanov homology with singular instanton knot homology.

• 15:00-16:00: Aliakbar Daemi (Simons Center)
  Chern-Simons functional and the homology cobordism group

  The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how one can construct a family of invariants of the homology cobordism group by applying ideas from min-max theory in Floer theory. The relationship between these invariants and the Froyshov's invariant will be discussed. I will also talk about some topological applications.

• 16:00-17:00: Tea

Thursday, January 4, 2018

• 07:30-08:30: Breakfast
• 08:30-09:30: Allison Miller (UT Austin)
  Knot traces and concordance

  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 surgeries. 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. I will also discuss the applicability of this work to the existence of interesting invertible satellite maps on the smooth concordance group.

• 09:45-10:45: Lisa Piccirillo (UT Austin)
  Integer knot surgeries and Stein knot traces

  It is well known that there exist pairs of knots K and K' in $S^3$ and integers n such that the two three-manifolds obtained by performing n-surgery on K and K' are homeomorphic, however the literature contains only a few systematic methods for constructing such a K and K'. This talk will begin with a survey of two of these methods. Define the n-trace of a knot K to be the four manifold obtained by attaching a single n framed two-handle to the four ball along K and observe that the boundary of the n-trace on K is n-surgery on K. It is natural then to ask when the discussed methods of constructing homeomorphisms of n-surgeries extend to diffeomorphisms of the n traces; I will discuss what is known about this. Finally I will discuss recent joint work with T. Mark and F. Vafaee which uses these ideas to show that certain knot traces which are not obviously Stein do in fact admit Stein structures.

• 11:00-12:00: John Etnyre (Georgia Tech)
  Contact surgeries and symplectic fillings

  It is well known that all contact manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. What is not so well understood is what properties of a contact structure are preserved by positive contact surgeries (the case for negative contact surgeries is fairly well understood now). In this talk we will discuss some new results about positive contact surgeries. In particular we will completely characterize when contact r surgery on a knot in the standard contact 3-sphere is symplectically fillable if r is in (0,1] and in general give obstructions to fillability in terms of the tau invariant of the knot. This is joint work with James Conway and Bulent Tosun.

• 12:00-13:45: Lunch break
• 13:45-14:45: Liam Watson (U de Sherbrooke)
  Bordered Floer homology via immersed curves, part 1

  The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. This pair of talks, which are part of a joint project with Hanselman, Rasmussen, and Watson, will set up this machinery and describe some of the applications that follow.

• 15:00-16:00: Jonathan Hanselman (Princeton U)
  Bordered Floer homology via immersed curves, part 2

  The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. This pair of talks, which are part of a joint project with Hanselman, Rasmussen, and Watson, will set up this machinery and describe some of the applications that follow.

• 16:00-17:00: Tea

Friday, January 5, 2018

• 07:00-08:00: Breakfast
• 08:00-09:00: Ian Zemke (Princeton U)
  Heegaard Floer mixed invariants of mapping tori

  The Seiberg Witten invariants of smooth closed 4-manifolds are powerful tools for studying 4-dimensional topology. The mixed invariants of 4-manifolds from Heegaard Floer homology are conjecturally equal to the Seiberg Witten invariants. In this talk we will describe how the graph TQFT for Heegaard Floer homology can be used to compute the Heegaard Floer mixed invariants of some 4-dimensional mapping tori in terms of Lefschetz numbers on ${\mathrm{HF}}^{+}$. The computation is in terms of traces, cotraces, and a "broken path" graph cobordism.

• 09:10-10:10: Akram Alishahi (Columbia U)
  Bordered Floer homology and compressibility

  In this talk, we will describe how bordered Floer homology detects homologically essential compressing disks and how bordered-sutured Floer homology detects partly boundary parallel tangles and bridges. Lipshitz-Ozsváth-Thurston have a factoring algorithm for computing bordered Floer homology. If time permits, we will describe an extension of their algorithm to compute bordered-sutured Floer homology. This is a joint work with Robert Lipshitz.

• 10:20-11:20: Robert Lipshitz (U of Oregon)
  Bordered ${\mathrm{HF}}^{-}$ with torus boundary: formal structure

  We will outline the formal structure of bordered ${\mathrm{HF}}^{-}$ with torus boundary and illustrate it with a few computations. This is joint work in progress with Peter Ozsváth and Dylan Thurston.

• 11:20-13:00: Lunch break
• 13:00-14:00: Adam Levine (Duke U)
  Concordance of knots in homology spheres

  Every knot in the 3-sphere bounds a non-locally flat piecewise-linear (PL) disk in the 4-ball, but Akbulut showed in 1990 that the same is not true for knots in the boundary of an arbitrary contractible 4-manifold. We strengthen this result by showing that there exists a knot K in a homology sphere Y (which is the boundary of a contractible 4-manifold) such that K does not bound a PL disk in any homology 4-ball bounded by Y. In more recent work (joint with Jen Hom and Tye Lidman), we show that the group of knots in homology spheres modulo non-locally-flat PL concordance is infinitely generated and contains an infinite cyclic subgroup.

• 14:10-15:10: Josh Greene (Boston College)
  Fibered simple knots

  John Luecke and I classified the simple knots in lens spaces that fiber. The answer takes an elementary number theoretic form, which I find satisfying if rather peculiar. I will focus on aspects of the proof that I have not elaborated on in previous talks.

• 15:10-16:00: Tea