# Past Topology Seminar

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.