# SEMINARI DI GEOMETRIA

## Rohlin’s theorem and non-simply connected 4-manifolds

Most applications of gauge theory in 4-dimensional topology are concerned with simply-connected manifolds with non-trivial second homology. I will discuss the opposite situation, first describing the classical Rohlin invariant for manifolds with first homology = Z and vanishing second homology. I will give an interpretation in terms of a Seiberg-Witten theory, with an unusual index-theoretic correction term. I will discuss recent work with Jianfeng Lin and Nikolai Saveliev giving a new formula for this invariant in terms of monopole homology.

## On corks with low shadow-complexity

A cork is a contractible Stein domain that gives arise to exotic pairs of 4-manifolds. The first example was found by Akbulut. It is known that any two exotic, simply-connected, closed 4-manifolds are related by a cork twist. We show that there are no corks having shadow-complexity zero. We also show that there are infinitely many corks having shadow-conplexity 1 and 2.

## Kodaira fibrations with small signature

ABSTRACT:

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration,

which are differentiable fibre bundles. First examples were constructed by

Kodaira and Atiyah to show that the signature need not be multiplicative in

fibre bundles.

We review approaches to construct such surfaces with focus on controlling the

signature. From this we obtain new examples with signature 16. (Joint work

with Ju A Lee and Michael Lönne)

## Elliptic diffeomorphisms of symplectic 4-manifolds

In his PhD thesis, Paul Seidel found examples of symplectomorphisms which are smoothly isotopic to the identity, but not isotopic to the identity within the symplectomorphism group.

## Entropy and length rigidity of negatively curved manifolds of finite volume

The problems of conjugacy rigidity and of entropy rigidity have a long history, with major breakthroughs in the last twenty years, for negatively curved compact manifolds and symmetric spaces. On the other hand, little is known for finite volume manifolds. I will survey the state of the art of these problems, and explain what can be generalized to finite volume manifolds, pointing out the main difficulties that arise in the non-compact case.

## Random groups and strong versions of Kazhdan’s Property (T)

Kazhdan's property (T) is a strong negation of amenability, relevant in the study of algebraic properties of groups, in the construction of expander graphs, in dynamics and in connection to the Baum-Connes conjectures. Various strengthened versions of property (T) have been formulated in recent years.

## New perspectives on lattice gauge field theories

## The universal quantum invariant and colored ideal triangulation

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, R.

## Torsion in negative curvature

A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We extend this theorem to the torsion part of the homology in all dimensions d > 3. From Gromov’s work it is known that in dimension 3 the size of torsion homology cannot be bounded in terms of the volume. In dimension 4 we give a somewhat precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d > 4 up to homeomorphism.