Schedule for: 16w5145 - Synchronizing Smooth and Topological 4-Manifolds

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will give an overview of topological concordance of knots and links. It will include disk embedding in dimension 4 and its relationship with concordance, filtrations defined from symmetric and asymmetric gropes and Whitney towers, invariants revealing their structures, and related topics as well as open questions in the area.

A complex line arrangement is a collection of complex projective lines in \(CP^2\) which may intersect at points of multiplicity greater than two. The combinatorial arrangements which can be geometrically realized and their space of realizations have been studied classically. We define symplectic, smooth, and topological versions of complex line arrangements in \(CP^2\), and study their realizability. While one might hope that these more flexible categories allow us to realize any combinatorics, in fact we show that there are obstructions to topological realizations of many combinatorial arrangements. Many open questions remain about realizability in different categories.

Meet in foyer of TCPL to participate in the BIRS group photo. Please don't be late, or you will not be in the official group photo! The photograph will be taken outdoors so a jacket might be required.

Knot Floer homology is a refinement of Heegaard Floer homology, providing an invariant for a pair (a 3-manifold, a knot in it). Recently, in collaboration with P. Ozsvath and Z. Szabo, we have found a 1-parameter 'deformation' of knot Floer homology for knots in S^3, leading to a family of concordance invariants. For the value t=1 the deformation admits a further symmetry, providing a bound on the unoriented 4 ball genus (smooth crosscap number) of the knot at hand. In the lecture I plan to recall the basic constructions behind knot Floer homology, list its basic properties, and show how the deformation works. Using grid diagrams we will discuss a sketch of the proof of the genus bounds.

In this talk I will discuss the basic properties of Pin(2)-monopole Floer homology (including some simple computational tools). This is the Morse-theoretic analogue of Manolescu's Pin(2)-equivariant Seiberg-Witten-Floer homology, and it can be used to provide an alternative disproof of the longstanding Triangulation Conjecture.

We use Ozsvath, Stipsicz, and Szabo's Upsilon invariant to provide bounds on cobordisms between knots that `contain full-twists'. This generalizes previous results and allows us to recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a full twist realize the braid index of their closure. We also provide inductive formulas for the Upsilon invariants of torus knots and compare the Upsilon function to the Levine-Tristram signature profile.

We will introduce the semigroup of L-space iterated torus knots. Then we will discuss the usage of semigroups and some subtleties in the computation of the Upsilon invariant for torus knots. Finally we will give some observations on the kernel of the Upsilon invariant.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In joint work with C. Manolescu, we use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to \(Z_4\)-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, 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.

Based on the notion of trisections of closed 4--manifolds of Gay and Kirby, Nick Castro and myself developed a rigorous definition of relative trisections as a generalization of trisections to 4--manifolds with boundary. In the talk I will present the basics of relative trisections starting with their relationship to open book decompositions of the bounding manifolds. I will then introduce a stabilization operation that gives rise to a statement about the uniqueness of relative trisections, thus complementing Gay and Kirby's proof of the existence of relative trisections. Finally, I will introduce the notion of diagrams of relative trisections and describe a method to recover the open book decomposition of the bounding manifold from the trisection diagrams. 

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold.  In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links - every such surface admits a decomposition into three standard pieces called a bridge trisection.  I'll describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface.  This is joint work with Alexander Zupan.

In 1981, Freedman proved that knots with trivial Alexander polynomial bound a locally flat disc in the four-ball. As a consequence, the degree of the Alexander polynomial constitutes an upper bound for the topological slice genus of a knot (F., 2015). We discuss a stronger bound, which is still determined solely by the knot's Seifert form. As sample applications, we will see upper bounds for the slice genus of torus knots (Baader, F., L., Liechti) and two-bridge knots (F., Mccoy), and for the stable slice genus of alternating knots (Baader, L.)."

it has been known for a long time that \(\p_1\)-null kernels imply surgery  ( pg 94 Freedman-Quinn book) and also that weaker grope based kernels are "universal" for surgery ( if you can solve these problems, you can solve all unobstructed surgery problems.) Slava and I have shown that a kind of kernel "half way between" the two is still universal.

The homology cobordism group of homology cylinders is enlargement of both the mapping class group and the concordance group of string links in homology \(D^2 \times I\) . We observe the structure of the group via filtration of extended Milnor invariants combined with Johnson homomorphisms. We also obtain deeper information invisible to previously known invariants by employing Hirzebruch-type intersection form defect invariants

The Heegaard Floer correction term (d-invariant) is an invariant of rational homology 3-spheres equipped with a \(\mbox{Spin}^c\) structure. In particular, the correction term of 1-surgeries along knots in the 3-sphere is a (2Z-valued) knot concordance invariant d1. In this work, we estimate d1 for the (2, q)-cable of any knot K. This estimate does not depend on the knot type of K. If K belongs to a certain class which contains all negative knots, then equality holds. By using this estimate, we obtain two corollaries. One of the corollaries shows that the relationship between d1 and the Heegaard Floer tau invariant is very weak in general. The other one gives infinitely many knots which cannot be unknotted either by only positive crossing changes or by only negative crossing changes.

4-dimensional surgery is a fundamental technique underlying geometric classification results for topological 4 manifolds. It is known to work in the topological category for a class of "good" fundamental groups. This result was originally established in the simply-connected case by Freedman in 1981, and it is currently known to hold for groups of sub exponential growth and a somewhat larger class generated by these. The A-B slice problem is a reformulation of the surgery conjecture for free groups, which is the most difficult case. In this talk I will show that the A-B slice problem admits a link-homotopy+ solution. The proof relies on geometric applications of the group-theoretic 2-Engel relation. I will also discuss implications for the surgery conjecture. (Joint work with Mike Freedman)

I plan to discuss two results related to 4-manifolds with boundary. The first, joint with Dave Auckly, Hee Jung Kim, and Paul Melvin, is a construction of diffeomorphisms of finite order on the boundary of certain contractible manifolds that change their smooth structure relative to the boundary. Tange has recently announced a similar result. We show in fact that for any finite group G acting on the 3-sphere, there is a G-action on the boundary of a contractible manifold, such that every element changes the smooth structure relative to the boundary. Our construction initially produces reducible boundaries, and then we show how to make these hyperbolic. The second set of results, joint with Arunima Ray, is concerned with two analogues of Dehn's lemma for 4-manifolds. We give examples of a reducible 3-manifold Y bounding a 4-manifold W that does not split smoothly as a boundary-connected sum, even though the reducing sphere in Y is null-homotopic in W. By a different construction, we find a contractible 4-manifold W with boundary a 3-manifold Y containing an essential torus that doesn't bound (smoothly, in one version; topologically in another version) a solid torus in W.

Joint with Taylor Martin, Carolyn Otto, and Jung Hwan Park. In the 1990's Cochran Orr and Teichner introduced a filtration of knot concordance indexed by half integers (the solvable filtration.)  Since then this filtration has been a convenient setting for many advances in knot concordance.  There are now many results in the literature demonstrating the difference between the n'th and (n.5)'th terms in this filtration, but none regarding the difference between the (n.5)'th and (n+1)'st.  In this talk we will prove that every genus one (0.5)-solvable knot is 1-solvable.  We will also provide a new sufficient condition for a high genus (0.5)-solvable knot to be 1-solvable and close with some possible candidates for knots which are (0.5)-solvable but not 1-solvable.

(Joint work with Lukas Lewark).  Rasmussen's invariant from perturbed Khovanov cohomology is a concordance homomorphism to the integers which also gives a lower bound on the smooth slice genus.  Khovanov-Rozansky sl(n) cohomology generalizes Khovanov cohomology (which appears as the case n=2) and perturbations of it give rise both to a slew of concordance homomorphisms which are also lower bounds as well as to lower bounds which are not equivalent to concordance homomorphisms.  For the case n=2 there is essentially only one perturbation, while already perturbations of the case n=3 exhibit complicated behavior.  We shall discuss this.

Gropes and Whitney towers are primary tools for studying 4-dimensional topology. As an effort to understand gropes and Whitney towers via the structure of their complements, we introduce notions of unknotted gropes/Whitney towers in 4-space. This is motivated by Freeman's result that an embedded 2-sphere in 4-space is topologically unknotted if its complement has infinite cyclic fundamental group. As an application, we establish grope and Whitney tower bi-filtrations of knots in 3-space by taking a slice of unknotted gropes/Whitney towers. Using the amenable signature theorem by Cha, which is based on the work of Cha and Orr, we prove that these bi-filtrations have rich structures. This is joint work with Jae Choon Cha.

A smooth knot slicing obstruction can be derived from Furuta's 10/8 theorem using 0-surgery on knots. I will show that this detects torsion elements in the concordance group and can be used to find topologically slice knots which are not smoothly slice. This is joint with F. Vafaee.

In the talk I will explain how to obtain lower bounds on unlinking numbers through 4-manifold techniques using a generalization of a theorem of Cochran-Lickorish. The method will be illustrated using links from Kohn's table whose unlinking numbers have only recently been determined through these methods.

In 2004 Peter Teichner and I stated a possible "if and only if criterion" for topological concordance to the unknot. I will recall that conjecture and I will give the evidence for the conjecture. I will also report on rather preliminary work with Patrick Orson on extending this conjecture to the concordance to the Hopf link. The audience should not expect any theorems. Lively questions and comments are warmly welcome.

Waldhausen's Theorem implies that any handle decomposition of the 3­-sphere can be simplified without introducing additional handles. The analogue in dimension 4 is unknown, but it is widely believed that there are handle decompositions of the standard smooth 4-­sphere which require additional pairs of canceling handles before they admit simplification. For handle decompositions without 1­-handles, it suffices to understand links in the 3-­sphere with certain Dehn surgeries ­­ these are the links characterized by the Generalized Property R Conjecture and its variations. We show that a given link has Stable Generalized Property R if and only if a certain infinite family of induced trisections is non­standard; thus, we provide evidence that trisections may be used to verify that the potential counterexamples introduced by Gompf-Scharlemann­-Thompson do not have Stable Generalized Property R. Parts of this talk are joint with Jeffrey Meier and Trenton Schirmer.

Every knot in the 3-sphere bounds a 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. The proof relies on using bordered Heegaard Floer homology to show that the action of a certain satellite operator on the knot concordance group is not surjective.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.