Baby Geometri
anno accademico 2023/2024

In this talk I will talk about a joint work with Rafael Torres, in which we create an exotic smooth structure on the total space of the non-trivial and non-orientable 2-sphere bundle over the real projective plane, where by “exotic smooth structure” we mean not equivalent to the standard one. Our procedure mimics Cappell-Shaneson’s construction of an exotic real projective 4-dimensional space, which I will sketch at the beginning of the talk as a warm up. The exotic structure we produce can also be obtained by applying a cut and paste procedure called Gluck twist to the total space of the trivial 2-sphere bundle over the real projective plane.

Two smoothly embedded surfaces in the same smooth 4-manifold are called exotic if they are C^0 ambient isotopic without being smoothly equivalent. In the talk, I will briefly introduce this peculiar phenomenon and show how it is possible to construct an infinite family of 2-knots (embedded spheres) and 2-links in a smooth 4-manifold that are pairwise exotic. Furthermore we will explore some of the properties of the constructed exotic families.

The talk is based on a joint work with Bais, Benyahia and Torres (see https://arxiv.org/abs/2206.09659). 

Given two mathematical objects, the most basic question we can ask is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a plan to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces. 

A classical result of Scott states that if G is the fundamental group of a compact surface and H<G is a finitely generated subgroup, then there is a finite-index subgroup G_1 that contains H as a retract (In this case, we say that G has "local retractions").

This property has been a powerful tool to verify the subgroup separability of fundamental groups of both manifolds and CW-complexes with some hierarchical structure (such as limit groups). We will discuss how in the latter (more combinatorial) setting, we can show a version of the local retractions property which, furthermore, preserves the L^2-homology of the initial group H. This is based on joint work with Sam Fisher. 

It is a well-known fact that the group of orientation-preserving isometries of the hyperbolic n-space is isomorphic to the matrix group SO^+(n,1). When n=2 and n=3, these groups have a "friendlier" description as the 2x2 matrix groups PSL(2,R) and PSL(2,C). 

By identifying R^4 with the quaternion algebra H, we will see that something similar happens in the n=5 case: more precisely, we will show that SO^+(5,1) is isomorphic to PSL(2,H) - the space of 2x2 quaternionic matrices with Dieudonné determinant equal to 1.

At the end of the talk, I will give an idea on how these results can be applied to try and understand deformations of complete hyperbolic 3-manifolds (with finite volume) in the 5-dimensional hyperbolic space. This is based on a joint work with Bruno Martelli.

The integral foliated simplicial volume (IFSV) of a manifold provides a dynamical version of its simplicial volume. In particular, it measures its fundamental cycles, parametrised by an action of the fundamental group on a probability space. The IFSV gives an upper bound for L2 Betti numbers and the cost of the fundamental group. In this talk, we provide a decomposition formula for the parametrised simplicial volume with respect to an ergodic decomposition of the action defining it, establishing an analogy with the ergodic decomposition formula for the cost of a group.

There have been various definitions of the Alexander invariants of a knot. Following some of these definitions one can generalise them so as to have coefficients twisted by a linear representation. The Alexander type invariants are known to detect non-trivial topological information(genus, hyperbolic volume of a knot etc). The twisted Alexander polynomial was introduced by Wada for knots and has been studied thereafter for more general manifolds as the complement of algebraic curves or line arrangements. We will discuss the relation of the twisted Alexander polynomial of the exterior manifold of a line arrangement and the twisted Alexander polynomial of its boundary manifold. We will present how using twisted Alexander polynomials induced by reducible representations, we can retract nontrivial topological information. 

We will also deal with the characteristic varieties of line arrangements, studied by various authors such as Zariski, Libgober, Artal. The main problem is to understand if the characteristic varieties are combinatorially determined in general. This is known to be true for their ”homogeneous part”, which corresponds to the resonance variety, as well as for the translated components having dimension at least one, as they are determined by orbifold pencils. This does not work in the same way for the 0-dimensional translated components. Here we present examples such that the characteristic variety has some translated 0-dimensional global component. 

Realizziamo quattro delle sei 3-varietà piatte, chiuse, orientabili come sezione delle cuspidi di una 4-varietà iperbolica orientabile, di volume finito, il cui gruppo di simmetrie agisce transitivamente sull'insieme delle cuspidi.

Stable commutator length is a measure of homological complexity of group elements. The goal of this talk is to explain some of its connections with negative curvature. We will discuss spectral gaps, and present a geometric proof of a theorem of Duncan and Howie that gives a sharp spectral gap in free groups. Along the way, we will encounter angle structures on 2-dimensional cell complexes and a combinatorial Gauß–Bonnet formula. 

Given two oriented, closed, connected manifolds, we can consider the set of mapping degrees of continuous maps between these manifolds. This raises the following question: Which subsets of Z can arise in this way? We present a construction by Neofytidis-Wang-Wang and Costoya-Munoz-Viruel that all finite subsets (containing 0) are mapping degree sets. Moreover, we will construct an explicit example of a subset that is not. 

In 1978, Hatcher and Thurston proposed a method to find a finite presentation of the mapping class group of a closed orientable surface. I will explain this method and outline how it can be used to find a finite presentation of the stabilizer of a spin structure on the surface. Along the way, I will also explain some applications to 4-manifolds.