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.

# Past Topology Seminar

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.

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

Elliptic cohomology is a family of generalised cohomology theories

$Ell_E^*$ parametrised by an elliptic curve $E$ (over some ring $R$).

Just like many other cohomology theories, elliptic cohomology admits

equivariant versions. In this talk, I will recall an old conjectural

description of elliptic cohomology, due to G. Segal, S. Stolz and P.

Teichner. I will explain how that conjectural description led me to

suspect that there should exist a generalisation of equivariant

elliptic cohomology, where the group of equivariance gets replaced by

a fusion category. Finally, I will construct $C$-equivariant elliptic

cohomology when $C$ is a fusion category, and $R$ is a ring of

characteristc zero.