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From Higher Invariants

Regensburg low-dimensional geometry and topology seminar

The seminar takes place on Zoom (see information below) every Tuesday at 16:00 CET (that is 7am on the US west coast, 10am on the US east coast, 10pm in China and 11pm in Korea and Japan).

Talks are split into a 40 minute and a 20 minute part, with a 15 minute tea break in between (you have to brew your own tea/coffee/beer...).

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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

Slides, Video

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 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.

The program is also contained in the RLGTS Google Calendar. To receive announcements by email, sign up to our mailing list.


Past talks in spring 2020

List of past talks in spring/summer 2020, with slides and videos

Zoom information

Zoom Meeting ID: 924 2829 6353

The password is the answer to the following riddle (lower case, singular, no article): As a topologist, what do you typically glue together from charts? (Hint: It's not an orbifold).

We encourage you to enable your video, for a more interactive atmosphere. The talks are recorded and made available on this webpage, including audio and video of questions from the audience.


Jonathan Bowden, Lukas Lewark, Raphael Zentner.