Giacomo Bascapè - Research

Research

I am interested in low-dimensional topology and SU(2)-representation theory, especially in 3-manifolds and knots, and links.

I am particularly interested in Problem 3.104 in Kirby's problem list: is it true that every 3-manifold admits a non-trivial SU(2)-representation of its fundamental group?

PhD Thesis

In my thesis, I classified all graph manifold rational homology 3-spheres with a single JSH torus that are SU(2)-abelian. I achieved this by defining an invariant, thereby avoiding the need for gauge theory.

Moreover, I proved that these SU(2)-abelian manifolds are Heegaard Floer L-space. This gives further evidence of the conjecture that SU(2)-abelian manifolds are Heegaard Floer L-spaces.

Papers

Toroidal graph manifolds with small homology are not SU(2)-abelian - To appear (2025)

SU(2)-abelian graph manifolds with a single JSJ torus - To appear in Algebraic & Geometric Topology (2025)

Older Projects

Reading Courses

With my colleague Patricia Sorya, I am the co-organizer with of an unofficial reading-course. In this reading course, which started in 2021, we studied the basics of Heegaard Floer homology theory and cyclic surgery theorem.

2022 GRADUATE STUDENT TOPOLOGY AND GEOMETRY CONFERENCE

I wrote a poster for the conference.

Wissenschaftliche Hilfskraft

In 2020 I was a research assistant (Wissenschaftliche Hilfskraft) in Regensburg for Raphael Zentner. During this time I studied the basis of Contact Topology, Confoliation and Foliation theory. I held a 4-days mini course whose aim was to go from the basic defition of the three theories to the famous article by Yakov Eliashber, which proved that "any weakly (resp. strongly) semi-fillable contact manifold is weakly (resp. strongly) fillable." Here you can find the notes of that mini course titled "A walk through Confoliation Garboil".