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.

# Past Topology Seminar

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.

The cobordism hypothesis gives a correspondence between the

framed local topological field theories with values in C and a fully

dualizable objects in C. Changing framing gives an O(n) action on the

space of local TFTs, and hence by the cobordism hypothesis it gives a

(homotopy coherent) action of O(n) on the space of fully dualizable

objects in C. One example of this phenomenon is that O(3) acts on the

space of fusion categories. In fact, O(3) acts on the larger space of

finite tensor categories. I'll describe this action explicitly and

discuss its relationship to the double dual, Radford's theorem,

pivotal structures, and spherical structures. This is part of work in

progress joint with Chris Douglas and Chris Schommer-Pries.

Given two actions of a group $G$ on trees $T_1,T_2$, Guirardel introduced the "core", a $G$--cocompact CAT(0) subspace of $T_1\times

T_2$. The covolume of the core is a natural notion of "intersection number" for the two tree actions (for example, if $G$ is a surface group

and $T_1,T_2$ are Bass-Serre trees associated to splittings along some curves, this "intersection number" is the one you'd expect). We

generalise this construction by considering a fixed finitely-presented group $G$ equipped with finitely many essential, cocompact actions on

CAT(0) cube complexes $X_1,...,X_d$. Inside $X=X_1\times ... \times X_d$, we find a $G$--invariant subcomplex $C$ which, although not convex

or necessarily CAT(0), has each component isometrically embedded with respect to the $\ell_1$ metric on $X$ (the key point is this change from

the CAT(0) to the $\ell_1$ viewpoint). In the case where $d=2$ and $X_1,X_2$ are simplicial trees, $C$ is the Guirardel core. Many

features of the Guirardel core generalise, and I will summarise these. For example, if the cubulations $G\to Aut(X_i)$ are "essentially

different", then $C$ is connected and $G$--cocompact. Time permitting, I will discuss an application, namely a new proof of Nielsen realisation

for finite subgroups of $Out(F_n)$. This talk is based on ongoing joint work with Henry Wilton.

Large-scale homology computations are often rendered tractable by discrete Morse theory. Every discrete Morse function on a given cell complex X produces a Morse chain complex whose chain groups are spanned by critical cells and whose homology is isomorphic to that of X. However, the space-level information is typically lost because very little is known about how critical cells are attached to each other. In this talk, we discretize a beautiful construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function.