Schedule for: 17w5040 - Low Dimensional Topology and Gauge Theory

We generalize the classical light bulb theorem to four dimensions. I.e. a smooth 2-sphere in $S^2\times S^2$ that intersects $S^2\times 0$ once and is homologous to $0\times S^2$ is smoothly isotopically standard.

We will present new constructions of small exotic symplectic 4-manifolds via rational blow-down surgery. This is a joint work with Anar Akhmedov.

The Euler characteristic is multiplicative in fiber bundles. On the other hand, the signature is not. Atiyah and, independently, Kodaira showed it by giving surface bundles over surfaces with non-zero signatures. Since then, many examples with non-zero signatures have been constructed. The signature of a surface bundle over a surface has some restrictions, for examples, it is dividable by 4 and vanishes if the base genus is 0 or 1. Bryan and Donagi constructed examples over a genus-2 surface with non-zero signatures. The signatures and the genera of their examples are sporadic. In this talk, for any positive integer n, we give a surface bundle of fiber genus g over a surface of genus 2 with signature 4n and a section of self-intersection 0 if g is greater than or equal to 39n. Such example are constructed using mapping class group arguments.

It is well known that for any exotic pair of simply connected closed 4-manifolds, one is obtained by twisting the other along a contractible submanifold. In contrast, we show that for each positive integer n, there exists an infinite family of pairwise exotic simply connected closed 4-manifolds such that, for any 4-manifold X and any compact (not necessarily connected) codimension zero submanifold W with boundary having first Betti number bounded by n, the family cannot be generated by twisting X along W and varying the gluing map. As a corollary, we show that there exists no `universal' 4-manifold with boundary generating all exotic families. Moreover, we give similar results for surgeries.

A necessary and sufficient condition for a smooth, compact 4-manifold to admit the structure of a Stein domain was given many years ago by Eliashberg, in terms of the existence of a certain kind of handle decomposition. In practice it is not always clear whether such a handle decomposition exists, even on a 4-manifold that is topologically very simple. We describe examples of 4-dimensional 2-handlebodies with a single 2-handle that do not obviously satisfy Eliashberg’s criteria yet still admit a Stein structure, and also examples of contractible 4-manifolds that do not admit any Stein structure despite satisfying various necessary conditions. The latter examples have bearing on a conjecture of Gompf asserting that no Brieskorn homology sphere admits a pseudoconvex embedding in complex 2-space.

We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.

A cork is a contractible smooth 4-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to detect exotic structures; in fact, any two smooth structures on a closed simply-connected 4-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup G of SO(4) there exists a contractible 4-manifold with an effective G-action on its boundary so that the twists associated to the non-trivial elements of G do not extend to diffeomorphisms of the entire manifold. In this talk, we will use Heegaard Floer techniques originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.

To every negative definite plumbing, Nemethi associates a computable combinatorial object called graded root. In the case the plumbing is the resolution of an almost rational singularity, the corresponding graded root captures the Heegaard Floer homology of the boundary 3-manifold. In this talk, I'll demonstrate how to detect the contact Ozsvath-Szabo invariant inside a graded root when the contact structure is compatible with an almost rational plumbing. As an application, we obstruct the existence of a Stein cobordism between the canonical contact structures on links of singularities. This is a joint work with F. Ozturk.

We shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. Our presentation is primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society), all joint with Thomas Leness.

The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of $\mathbb{R}P^1$) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.

Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K), which they also showed is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain a collection of τ-invariants, one for each spin-c structure. In addition, these invariants can be used to obtain a lower bound on the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y.

We construct an invariant of closed spin^c 4-manifolds. This invariant is defined using families of Seiberg-Witten equations and formulated as a cohomology class on a certain abstract simplicial complex. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.

It was proved in the 1990's by Curtis-Freedman-Hsiang-Stong and Matveyev that any two homeomorphic, closed, simply-connected smooth 4-manifolds are related by removing and regluing a single compact contractible submanifold, called a cork. This talk will present joint work with Paul Melvin which generalizes this theorem to any finite list of homeomorphic, closed, simply-connected, smooth 4-manifolds. We will then apply a relative version of this finite order result to address infinite lists of homeomorphic, smooth 4-manifolds. Although in this case, a strictly analogous theorem is not possible, as noted by Tange and Yasui, extensions can be obtained by relaxing the compactness condition on the cork.

It has been known for over three decades that $\mathbb{R}^4$ has uncountably many exotic smoothings, exhibiting a failure of existence of diffeomorphisms. This talk discusses the first results on the corresponding uniqueness problem: Using a trick from cork theory, we will exhibit exotic $\mathbb{R}^4$s with uncountably many isotopy classes of self-diffeomorphisms. We will obtain many explicit group actions injecting into the diffeotopy group, including examples contrasting sharply with Taylor's results on isometry groups. We will also obtain infinite group actions at infinity for which no nontrivial element extends over the whole exotic $\mathbb{R}^4$, contrasting with another exotic $\mathbb{R}^4$ for which every diffeomorphism at infinity extends. Details appear in a recent arXiv preprint.

A classical conjecture of Akbulut and Kirby asserted that if a pair of knots have homeomorphic 0-surgeries then the knots should be (smoothly) concordant. This was disproven in 2015 by Yasui; his proof used a concordance invariant which is also a diffeomorphism invariant of the four manifold ' trace' of the knot surgery. This led Abe to assert a corrected conjecture; if a pair of knots have diffeomorphic 0-surgery traces then the knots should be concordant. We give a method for constructing many pairs of distinct knots with diffeomorphic 0-surgery traces and use the d-invariants of Heegaard Floer homology to obstruct the smooth concordance of some of these knots, thereby disproving Abe's conjecture. As a consequence, we obtain a proof that there exist interesting bijective maps on the smooth concordance group coming from the satellite construction. This is joint work with Allison Miller.

Virtual knot theory concerns knots in thickened surfaces, and Turaev introduced virtual concordance and several useful invariants of them. This talk is based on joint work in progress with Micah Chrisman and Robin Gaudreau, and our goal is to extend various classical concordance invariants to the virtual setting and apply them to determine the sliceness and the 4-genus for low crossing virtual knots. One of the obstacles in virtual knot theory is the absence of Seifert surfaces, and for that reason we focus on the subclass of virtual knots with homologically trivial representatives. These knots admit Seifert surfaces, and we use them to define the usual package of knot invariants, including Alexander-Conway polynomials, signatures, and twisted signatures. In general, the resulting invariants depend on the choice of Seifert surface, and they often (but not always) give rise to concordance invariants of long virtual knots. The untwisted signatures can be computed in terms of Goeritz matrices a la Gordon-Litherland and using Manturov projection, signature invariants can be extended from the nomologically trivial knots to all virtual knots. We apply these and other invariants to determine sliceness for virtual knots with up to 6 crossings.

I will construct infinite families of Rohlin invariant one Brieskorn homology spheres bounding rational balls (a joint work with Kyle Larson). I will then switch over discussing homology spheres bounding contractible manifolds. A contractible manifold W can fail to be a cork, either if there is no exotic involution on its boundary, or W fails to be Stein. We will give examples of Stein non-corks, and non-Stein non-loose corks. It is still an open question whether there are loose-corks that can not be corks (this part is a joint work with Danny Ruberman).

The original Thom conjecture states that holomorphic curves are minimal genus representatives of 2-dimensional homology classes in $\mathbb{C}P^2$. It has been known for a long time that the analogous claim for codimension 2 homology classes in $\mathbb{C}P^n$ does not hold; Freedman showed that for n even any such class is represented by a submanifold which has smaller middle homology than a complex hypersurface representing this class and which on the level of homotopy behaves as a complex hypersurface. We consider the case of 4-manifolds in $\mathbb{C}P^3$ and show that the rank of the 2nd homology in any given class can be significantly reduced. This is joint work with D. Ruberman and M. Slapar.

We introduce the Kodaira dimension of contact 3-manifolds and show that contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. We also prove that, given any contact 3-manifold, there is a lower bound of $2\ chi+ 3\ sigma$ for all its minimal symplectic fillings. This generalizes the similar bound of Stipsicz for Stein fillings. This talk is based on joint works with Cheuk Yu Mak, and partly with Koichi Yasui.

The Wall's stable h-cobordism theorem states that homotopy equivalent, smooth simply-connected 4-manifolds become diffeomorphic after stabilizing, i.e. connected summing with some finite number of a S^2-bundle over S^2. And, in fact, all known examples need only one stabilization to be diffeomorphic. In this talk, we will talk about the analogous stabilization question for knotted surfaces in simply-connected 4-manifolds produced by all of the known constructions based on Fintushel-Stern knot surgery. And we will prove that any pair of these knotted surfaces that preserve the fundamental groups of their complements become all diffeomorphic after single stabilization.