Past Topology Seminar

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 RP^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.

I will talk about the diffeomorphism classification of 4-manifolds up to 
connected sums with the complex projective plane, and how the resulting 
equivalence class of a manifold can be detected by algebraic topological 
invariants of the manifold.  I may also discuss related results when one 
takes connected sums with another favourite 4-manifold, S^2 x S^2, instead.

I will report on a joint project with Frank Swenton whose goal is to develop an algorithm to determine whether an alternating knot is ribbon.  We can’t do this yet but we have an algorithm that has been remarkably, and indeed mysteriously, successful in finding a great deal of new slice knots.

The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of M. We associate the analog of a Lie group G(m) to the Monster Lie algebra m. We give generators for large free subgroups and we describe relations in G(m).

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants as envisioned by Deninger.

Fix a loop group LG, a level k∈ℕ, and let Repᵏ(LG) be corresponding category of positive energy representations.
For any pair of pants Σ (with complex structure in the interior and parametrized boundary), there is an associated functor Repᵏ(LG) × Repᵏ(LG) → Repᵏ(LG): (H,K) ↦ H⊠K, called the fusion product.

It had been widely expected (but never proven) that this operation should be unitary. Namely, that the choice of LG-invariant inner products on H and on K should induce an LG-invariant inner product on H⊠K. We show that this is not the case: there is an anomaly.
More precisely, there is an ℝ₊-torsor canonically associated to Σ. It is only after trivialising of this ℝ₊-torsor that the fusion product acquires an LG-invariant inner product. (The same statement applies when Σ is an arbitrary Riemann surface with boundary.)
Joint work with James Tener.

In this talk I will construct a reduced tensor product of braided fusion categories containing a symmetric fusion category $\mathcal{A}$. This tensor product takes into account the relative braiding with respect to objects of $\mathcal{A}$ in these braided fusion categories. The resulting category is again a braided fusion category containing $\mathcal{A}$. This tensor product is inspired by the tensor product of $G$-equivariant once-extended three-dimensional quantum field theories, for a finite group $G$.
To define this reduced tensor product, we equip the Drinfeld centre $\mathcal{Z}(\mathcal{A})$ of the symmetric fusion category $\mathcal{A}$ with an unusual tensor product, making $\mathcal{Z}(\mathcal{A})$ into a 2-fold monoidal category. Using this 2-fold structure, we introduce a new type of category enriched over the Drinfeld centre to capture the braiding behaviour with respect to $\mathcal{A}$ in the braided fusion categories, and use this encoding to define the reduced tensor product.
 

I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

Let \Gamma_{g,1} denote the mapping class group of a genus g surface with one parametrized boundary component.  The group homology H_i(\Gamma_{g,1}) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g > 3i.  Outside that range, the relative homology groups H_i(\Gamma_{g,1},\Gamma_{g-1,1}) contain interesting information about the failure of homological stability.  In this talk, I will discuss a metastability result; the relative groups depend only on the number k = 2g-3i, as long as g is large compared to k.  This is joint work with Alexander Kupers and Oscar Randal-Williams.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

Boundaries of hyperbolic spaces have played a key role in low dimensional topology and geometric group theory.  In 1993, Paulin showed that the topology of the boundary of a (Gromov) hyperbolic space, together with its quasi-mobius structure, determines the space up to quasi-isometry.  One can define an analogous boundary, called the Morse boundary, for any proper geodesic metric space.  I will discuss an analogue of Paulin’s theorem for Morse boundaries of CAT(0) spaces. (Joint work with Devin Murray.)

We will discuss a construction of cobordism maps on the full link complex for decorated link cobordisms. We will focus on some formal properties, such as grading change formulas and local relations. We will see how several expressions for mapping class group actions can be interpreted in terms of pictorial relations on decorated surfaces. Similarly, we will see how these pictorial relations give a "connected sum formula" for the involutive concordance invariants of Hendricks and Manolescu.

After giving an introduction to functorial field theories I will explain a natural generalization thereof, called "twisted" field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to Freed-Teleman's "relative" boundary field theories. Finally, I will explain some examples, one of which explicitly arises from factorization homology and whose target is the higher Morita category of E_n-algebras, bimodules, bimodules of bimodules etc.

An efficient way to descibe binary operations which are associative only up to coherent homotopy is via simplicial spaces. 2-Segal spaces were introduced independently by Dyckerhoff--Kapranov and G\'alvez-Carrillo--Kock--Tonks to encode spaces carrying multivalued, coherently associative products. For example, the Waldhausen S-construction of an abelian category is a 2-Segal space. It describes a multivalued product on the space of objects given in terms of short exact sequences. 
The main motivation to study spaces carrying multivalued products is that they can be linearised, producing algebras in the usual sense of the word. For the preceding example, the linearisation yields the Hall algebra of the abelian category. One can also extract tensor categories using a categorical linearisation procedure.
In this talk I will discuss double 2-Segal spaces, that is, bisimplicial spaces which satisfy the 2-Segal condition in each variable. Such bisimplicial spaces give rise to multivalued bialgebras. The second iteration of the Waldhausen S-construction is a double 2-Segal space whose linearisation is the bialgebra structure given by Green's Theorem. The categorial linearisation produces categorifications of Zelevinsky's positive, self-adjoint Hopf algebras.
 

Knots and links naturally appear in the neighbourhood of the singularity of a complex curve; this creates a bridge between algebraic geometry and differential topology. I will discuss a topological approach to the study of 1-parameter families of singular curves, using correction terms in Heegaard Floer homology. This is joint work with József Bodnár and Daniele Celoria.

The technique of scanning, or the parameterised Pontrjagin--Thom construction, has been extraordinarily successful in calculating the cohomology of configuration spaces (McDuff), moduli spaces of Riemann surfaces (Madsen, Tillmann, Weiss), moduli spaces of graphs (Galatius), and moduli spaces of manifolds of higher dimension (Galatius, R-W, Botvinnik, Perlmutter), with constant coefficients. In each case the method also works to study the cohomology of moduli spaces of objects equipped with a "tangential structure". I will explain how choosing an auxiliary highly-symmetric tangential structure often lets one calculate the cohomology of these moduli spaces with large families of twisted coefficients, by exploiting the symmetries of the tangential structure and using a little representation theory.

 

 Markus Szymik and I computed the homology of the Higman-Thompson groups by first showing that they stabilize (with slope 0), and then computing the stable homology. I will in this talk give a new point of view on the computation of the stable homology using Thumann's "operad groups". I will also give an idea of how scanning methods can enter the picture. (This is partially joint work with Søren Galatius.) 

It is hard to detect the exotic nature of an exotic n-sphere M 
in homotopical features of the diffeomorphism group Diff(M). The well 
known reason is that Diff(M) contains a big topological subgroup H which 
is identified with the group of diffeomorphisms rel boundary of the 
n-disk, with a small coset space Diff(M)/H which is invariably homotopy 
equivalent to O(n+1). Therefore it seems that our only chance to detect 
the exotic nature of M in homotopical features of Diff(M) is to see 
something in this extension.  (To make sense of "homotopical features of 
Diff(M)" one should think of Diff(M) as a space with a multiplication 
acting on an n-sphere.) I am planning to report on PhD work of O Sommer 
and calculations due to myself and Sommer which, if all goes well, would 
show that Diff(M) has some exotic homotopical properties in the case 
where M is the 7-dimensional exotic sphere of Kervaire-Milnor fame which 
bounds a compact smooth framed 8-manifold of signature 8. The 
theoretical work is based on classical smoothing theory and the 
calculations would be based on ever-ongoing (>30 years) joint work 
Weiss-Williams, and might give me and Williams another valuable 
incentive to finish it.

We construct the diffeomorphism-equivariant “scanning map” associated to the configuration spaces of manifolds with twisted summable labels. The scanning map is also functorial with respect to embeddings of manifolds. To adapt P. Salvatore's idea of non-commutative summation into twisted setting, we define a bundle of Fulton-MacPherson operads over a manifold M whose fibres are built within tangent spaces of M.

In 1956 Milnor published a paper proving that there are manifolds homeomorphic to the 7-sphere but not diffeomorphic to it. Seeking to generalise this example, he was led in around 1960 to introduce a construction for  killing homotopy groups of manifolds. When this was generalised to killing relative homotopy groups it became a general and powerful method of construction. An obstruction arises to killing the last group, and the analysis of this obstruction in general leads to a new theory.

Any closed 3-manifold can be obtained by glueing two handle bodies along their boundary. For a fixed such glueing, any other differs by changing the glueing map by an element in the mapping class group. Beginning with an idea of Dunfield and Thurston, we can use a random walk on the mapping class group to construct random 3-manifolds. I will report on recent work on the structure of such manifolds, in particular in view of tower of coverings and their topological growth: Torsion homology growth, the minimal degree of a cover with positive Betti number, expander families. I will in particularly explain the connection to some open questions about the mapping class group.