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Past talks of the Regensburg low-dimensional geometry and topology seminar

Link to the talks during the current semester.

Talks in fall/winter 2020/21

Date	Speaker	Title	Abstract
November 3	Marco Marengon (MPIM Bonn)	Non-orientable knot cobordisms and torsion order in Floer homologies Video Slides	Given an orientable knot cobordism between classical knots K and K', Juhász, Miller, and Zemke proved an inequality involving the torsion order of the knot Floer homology of K and K' and the Euler characteristic and the number of local maxima appearing in the cobordism. In our work, we prove analogous inequalities for unorientable knot cobordisms, using unoriented versions of instanton and knot Floer homology. Much of the subtlety in our argument lies in the fact that we need to choose more complicated decorations on the surface than the ones appearing in Juhász-Miller-Zemke's case. We also introduce unoriented versions of the band unknotting number and of the refined cobordism distance and apply our results to give bounds on these based on the torsion orders of the Floer homologies. As an application, I will exhibit a family of knots all of which bound an embedded Möbius band in B^4, but for which the unorientable refined cobordism distance from the unknot is arbitrarily large. This is joint work with Sherry Gong.
November 10	Masaki Taniguchi (Riken)	Filtered instanton Floer homology and the 3-dimensional homology cobordism group Video Slides	We introduce a family of real-valued homology cobordism invariants r_s(Y) of oriented homology 3-spheres. The invariants r_s(Y) are based on a quantitative construction of filtered instanton Floer homology. Using our invariants, we give several new constraints of the set of smooth boundings of homology 3-spheres. As one of the corollaries, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. As another corollary, we show that if the 1-surgery of a knot has negative Froyshov invariant, then the 1/n-surgeries (n>0) of the knot are linearly independent in the homology cobordism group. This is joint work with Yuta Nozaki and Kouki Sato.
November 17	Louis-Hadrien Robert (Luxembourg)	Categorification of 1 and of the Alexander polynomial Video Slides	I'll give a combinatorial and down-to-earth definition of the symmetric gl(1) homology. It is a (non-trivial) link homology which categorifies the trivial link invariant (equal to 1 on every link). Then I'll explain how to use this construction to obtain colored categorification of the Alexander polynomial. (joint with E. Wagner)
November 24	Sarah Rasmussen (Cambridge)	Taut foliations from left orders in Heegaard genus 2 Video Slides	Until now, taut foliations on non-fibered hyperbolic 3-manifolds have generally been constructed using branched surfaces, whether via sutured manifold hierarchies, spanning surfaces of knot exteriors, or Dunfield's one-vertex triangulations with foliar orientations. In this talk, I introduce a novel taut foliation construction that makes no recourse to branched surfaces. Instead, it starts with a transversely-foliated $\mathbb{R}$-bundle over a Heegaard surface, specified by a real line action from a left-invariant order (when existent) on the fundamental group of the 3-manifold. Relating taut foliations to fundamental group real line actions is a decades old question that interested Thurston, Gabai, and Calegari, and has been revived in recent years by the L-space conjecture. This construction works reliably for genus 2 Heegaard diagrams satisfying mild conditions, explaining certain numerical results of Dunfield.
December 1	Tao Li (Boston College)	Heegaard genus, degree-one maps, and amalgamation of 3-manifolds Video Slides	Let $W$ be the exterior of a knot in a homology sphere and let $M$ be an amalgamation of $W$ and any other compact 3-manifold along boundary torus. Let $N$ be the manifold obtained by pinching $W$ into a solid torus. This means that there is a degree-one map from $M$ to $N$. We prove that the Heegaard genus of $M$ is at least as large as the Heegaard genus of $N$. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.
December 8	Ciprian Manolescu (Stanford)	Relative genus bounds in indefinite four-manifolds Video Slides	Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X - B^4, with boundary a knot K. We give several methods to produce bounds on the genus of such surfaces in a fixed homology class. Our techniques include relative adjunction inequalities and the 10/8 + 4 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold. We give an example showing that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. Further, we give examples of knots that are topologically but not smoothly H-slice in some indefinite 4-manifolds. This is joint work with Marco Marengon and Lisa Piccirillo.
December 15	Siddhi Krishna (Georgia Tech)	Taut foliations and Dehn surgery along positive braid knots Video Slides	The L-space conjecture has been in the news a lot lately. It predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3-manifold Y. In particular, it predicts exactly which 3-manifolds admit a "taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections. In particular, I'll discuss a strategy for building taut foliations manifolds obtained by Dehn surgery along knots realized as closures of "positive braids". As an application, I will show how taut foliations can be used to obstruct positivity for cable knots. All are welcome; no background in foliation or Floer homology theories will be assumed.
January 19	Paul Feehan (Rutgers)	Morse-Bott theory on analytic spaces and applications to topology of smooth 4-manifolds Video Slides	We describe a new approach to Morse theory on singular analytic spaces of the kind that typically arise in gauge theory, such as the moduli space of SO(3) monopoles over 4-manifolds or the moduli space of Higgs pairs over Riemann surfaces. We explain how this new version of Morse theory, called virtual Morse-Bott theory, can potentially be used to answer questions arising in the geography of 4-manifolds, such as whether constraints on the topology of compact complex surfaces of general type continue to hold for symplectic 4-manifolds or even for smooth 4-manifolds of Seiberg-Witten simple type. This is joint work with Tom Leness.
January 26	Danny Ruberman (Brandeis)	The diffeomorphism group of a 4-manifold Video Slides	Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group. The lowest dimension for such classical phenomena is 5. I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)-parameter gauge theory. The construction uses a topological technique.
February 2	Luisa Paoluzzi (University of Aix-Marseille)	Cyclic branched covers of alternating knots Video Slides	The goal of the talk is to show that certain topological invariants of knots, called cyclic branched covers, are strong invariants for prime alternating knots. I will start by describing what cyclic branched covers are before presenting some known facts concerning their properties as invariants of knots. I will then point out what features of alternating knots are key to prove the result. The actual proof will be given in the second part of the talk.
February 9	Liam Watson (University of British Columbia)	Symmetry and mutation	Mutation is a relatively simple process for altering a knot in a non-trivial way, but it turns out to be quite tricky to see the difference between mutant pairs—a surprisingly wide range of knot invariants are unable to distinguish mutants. In the first part of the talk, I will give some background on the symmetry group associated with a knot, and show that this group is sometimes able to see mutation. In the second part of the talk, I will outline some work with Andrew Lobb, in which we appeal to a symmetry—when present—in order to define a refinement of Khovanov homology that is able to separate mutants.
February 16	Nick Salter (Columbia)	r-spin mapping class groups and applications Video Slides	An r-spin structure on a surface can be thought of as a gadget for measuring “Z/rZ-valued winding numbers” of curves on the surface. There is an evident action of the mapping class group on the set of such objects; an r-spin mapping class group is the associated stabilizer. r-spin structures appear in a wide variety of contexts at the interface of topology and algebraic geometry: singularity theory, translation surfaces/Abelian differentials, linear systems on algebraic surfaces. I will explain what is now known about r-spin mapping class groups and the uses to which the theory can be put. This encompasses various projects with my collaborators Aaron Calderon and Pablo Portilla Cuadrado.

Talks in spring/summer 2020

Date	Speaker	Title	Downloads	Abstract
May 5	Peter Kronheimer (Harvard)	Genus versus double-points for immersed surfaces	Video	If X is a simply-connected closed 4-manifold containing an oriented embedded surface S of genus g, is there always an immersed sphere S' which represents the same homology class and has only g transverse double-points? Colloquially, can we "trade handles for double points"? This is an open question, though a "relative" version of the question (concerning surfaces in the 4-ball bounding a given knot in the 3-sphere) is known to have a negative answer. For closed surfaces in closed 4-manifolds, a particularly interesting class of examples comes from algebraic geometry, and includes the question of whether two smooth quintic surfaces can intersect in a singular rational curve. We will explore whether gauge theory might be a tool that can be used to explore these questions.
May 12	Danny Calegari (University of Chicago)	Taut foliations leafwise branch cover the 2-sphere	Video Slides	A co-oriented foliation of an oriented 3-manifold is taut if and only if there is a map from the 3-manifold to the 2-sphere which is a branched cover when restricted to each leaf. I shall give two proofs of this theorem and explain its relation to theorems of Ghys and Donaldson.
May 19	Richard Webb (University of Manchester)	Quasimorphisms on diffeomorphism groups	Video Slides	I will explain how to construct an unbounded quasimorphism on the group of isotopically-trivial diffeomorphisms of a surface of positive genus. As a corollary the commutator length and fragmentation norm are both (stably) unbounded, which solves a problem of Burago--Ivanov--Polterovich. The proof uses a new hyperbolic graph on which these groups act by isometries, which is inspired by techniques from mapping class groups. This is joint work with Jonathan Bowden and Sebastian Hensel.
May 26	Zhenkun Li (MIT)	Instanton Floer homology and the depth of taut foliations	Video Slides	Sutured manifold hierarchy is a powerful tool introduced by Gabai to study the topology of 3-manifolds. The length of a sutured manifold hierarchy gives us a measurement of how complicated the sutured manifold is. Also, using this tool, Gabai proved the existence of finite depth taut foliations. However, he didn’t discuss how finite the depth could be. Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka and is defined on balanced sutured manifolds. In this talk, I will explain how sutured instanton Floer homology could offer us bounds for the minimal length of a sutured hierarchy and the minimal depth of a foliation on a fixed balanced sutured manifold.
June 2	no talk (pentecoste)
June 9	Boyu Zhang (Princeton)	Instanton Floer homology for tangles and applications in Khovanov homology	Video Slides	In this talk, I will present an excision formula for singular instanton Floer homology where the excision surfaces intersect the singular locus. This formula allows us to define an instanton Floer homology theory for sutured manifolds with tangles. Similar to the non-singular case, the instant Floer homology for tangles is non-vanishing on taut manifolds with tangles and detects trivial products. As applications, we prove several detection results for Khovanov homology. This is joint work with Yi Xie.
June 16	Gordana Matic (Univ. of Georgia/MPIM Bonn)	Spectral order invariant and obstruction to Stein fillability	Video Slides	In this joint work with Cagatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand, we define an invariant of contact structures in dimension three arising from introducing a filtration on the boundary operator in Heegaard Floer homology. This invariant takes values in the set of positive integers union infinity. It is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. I will give the definition and discuss computability of the invariant. As an application, we give an easily computable obstruction to Stein fillability on closed contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
June 23	Lisa Piccirillo (MIT)	The trace embedding lemma and PL surfaces	Video Slides	4-manifold topologists have long been interested in understanding smooth (resp. topological) embedded surfaces in smooth (resp. topological) 4-manifolds, and as such have developed rich suites of tools for obstructing the existence of smooth (resp. topological) surfaces. Understanding PL surfaces in smooth 4-manifolds has historically garnered less interest, but several problems about PL surfaces have recently arisen in modern lines of questioning. Presently there are far fewer tools available to obstruct PL surfaces. In this talk, I’ll discuss how to use a classical observation, called the trace embedding lemma, to repurpose smooth surface obstructions as PL surface obstructions. I’ll discuss applications of these retooled obstructions to problems about spinelessness, exotica, and geometrically simply connectedness. This is joint work with Kyle Hayden.
June 30	Kathryn Mann (Cornell)	Large-scale geometry of big mapping class groups	Video Slides	Mapping class groups of infinite type surfaces are not finitely generated (nor even are they locally compact) groups, but in many cases one can still meaningfully talk about their large scale geometry. I will explain joint work with Kasra Rafi on the problem of which surfaces have mapping class groups with nontrivial coarse geometry, and how this relates to questions of actions on arc and curve complexes.
July 7	Mikhail Khovanov (Columbia University)	From SL(2) to GL(N) foam evaluation	Video Slides	Foams in 3-space are cobordisms between planar graphs that are heavily used in link homology theories. In this talk we'll explain how foam theory can be built from the ground up starting with an evaluation of closed foams, for SL(2) foams, then GL(2) foams, and finally GL(N) foams for any N. The talk is based on joint work with Louis-Hadrien Robert and on L.-H. Robert and Emmanuel Wagner's work on foam evaluation.
July 14	Claudius Zibrowius (UBC Vancouver)	Thin links and Conway spheres	Video Slides	When does Dehn surgery along a knot give an L-space? More generally, when does splicing two knot complements give an L-space? Hanselman, Rasmussen and Watson gave very compelling answers to these questions using their technology of immersed curves for three-manifolds with torus boundary. Similar invariants have been developed for four-ended tangles. We use those invariants to study various notions of thinness in both Heegaard Floer and Khovanov homology from the perspective of tangle decompositions along Conway spheres. Interestingly, our results bear strong resemblance to the aforementioned results about L-spaces. Also, we observe strong similarities between Heegaard Floer and Khovanov homology that lead us to ask: What is a thin link? This is joint work with Artem Kotelskiy and Liam Watson.
July 21	Marc Lackenby (University of Oxford)	Knot genus in a fixed 3-manifold	Video Slides	The genus of a knot is the minimal genus of any of its Seifert surfaces. This is a fundamental measure of a knot's complexity. It generalises naturally to homologically trivial knots in an arbitrary 3-manifold. Agol, Hass and Thurston showed that the problem of determining the genus of a knot in a 3-manifold is hard. More specifically, the problem of showing that the genus is at most some integer g is NP-complete. Hence, the problem of showing that the genus is exactly some integer g is not in NP, assuming a standard conjecture in complexity theory. On the other hand, I proved that the problem of determining the genus of a knot in the 3-sphere is in NP. In my talk, I will discuss the problem of determining knot genus in a fixed 3-manifold. I will outline why this problem is also in NP, which is joint work with Mehdi Yazdi. The proof involves the computation of the Thurston norm ball for knot exteriors.