### Geometry and topology of manifolds

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget: €90.750,00

Principal Investigator: Bruno Martelli (Università di Pisa)

##### Participants

Paolo Lisca (Università di Pisa), Carlo Petronio (Università di Pisa), Giuseppe Bargagnati (Università di Pisa), Filippo Bianchi (Università di Pisa), Carlo Collari (Università di Pisa), Roberto Frigerio (Università di Pisa), Domenico Marasco (Università di Pisa), Alice Merz (Università di Pisa), Andrea Parma (Università di Pisa), Andrea Tamburelli (Università di Pisa)

##### Description

The project aims at studying the geometry and topology of smooth manifolds. Manifolds have been a central object of research in the last 150 years, and the study of the interplay between their geometry and topology is of primary importance. We subdivide our project in two areas, with large overlaps.
Geometry:
A central theme in the study of manifolds is the assignment and study of a geometric structure. The prominent role is played by hyperbolic geometry, but also various other structures may arise. We single out these directions of research:
a) Fibrations on hyperbolic manifolds of higher dimension;
b) Deformations and rigidity of hyperbolic structures;
c) Study of submanifolds;
d) Complex hyperbolic manifolds, $H^{p,q}$ structures;
e) Geometric group theory: Culler-Vogtmann Outer Space of free groups;
f) Dynamics on moduli spaces;
g) Circle packings.
Shortly, we aim to:
a) Construct and study fibrations in higher dimension;
b) Study deformations and rigidity fenomena in dimension 3 and 4;
c) Study geodesic immersions in the arithmetic and non-arithmetic setting;
d) Study reflection groups and their deformations, and maximal embeddings in $H^{p,q}$;
e) Solve decision problems as conjugacy and reducibility for automorphisms of free products;
f) Study dynamics of flows on translation surfaces, billiards, fractals arising in dynamical settings;
g) Rigidity of circle packings on projective surfaces of hyperbolic type.
Topology:
We focus mostly on the topology of manifolds 2, 3, and 4. We single out these directions of research:
1) Special handlebody decompositions of smooth 4-manifolds and non-symplectic embeddings of rational homology 4-balls;
2) The Hurwitz existence problem for ramified covers between surfaces;
3) Apparent contours of surfaces on flow-spines of 3-manifolds;
4) Heegaard Floer homology, taut foliations and left-orderable groups;
5) Poset homology and its applications in topology;
6) Strongly invertible (Legendrian) links;
7) Quasimorphisms and braid groups;
8) Bounded cohomology and simplicial volume.
Shortly, we aim to:
1) Understand some handlebody decompositions of smooth 4-manifolds and investigate their applications;
2) Solve unsettled instances for length-2 partitions and more geometrically motivated ones;
3) Develop reconstruction algorithms and combinatorial moves translating isotopy;
4) Prove (or disprove) the L-space conjecture for large families of 3-manifolds;
5) Investigate the properties of poset homologies and their applications in algebraic and low-dimensional topology;
6) Define new invariants of strongly invertible links and strongly invertible Legendrian links and find potential applications;
7) Define and study new quasimorphisms on braid groups and their subgroups and generalizations of Levine-Tristram signatures for classical links;
8) Better understand geometric invariants via new computations.