• (co)homology of arrangements
• asphericity
• hyperplane arrangements
• local systems

This thesis, supervised by Professor Filippo Gianluca Callegaro, addresses the problem of hyperplane arrangements that are not K(pi,1). I have analysed the following works:
- The papers by Hattori (1976) and Salvetti (1987) on the non-asphericity of arrangements in general position.
- The 2002 paper by Papadima and Suciu, in which Hattori’s results are extended to the class of hypersolvable arrangements, for which asphericity is shown to be equivalent to being supersolvable.
- A recent 2024 paper by Yoshinaga, which introduces a new method for constructing homotopically non-trivial spheres in the complement of an arrangement. Starting from a real arrangement, the paper introduces the notion of consistent and locally consistent systems of open half-spaces. A sphere
S(e) is then constructed, immersed in the complement of the complexified arrangement M(A), and using a twisted version of the intersection number, it is shown that this sphere is homotopically trivial if and only if the system of half-spaces is globally consistent.

As part of my original research work, I found a way to construct Yoshinaga’s homotopically non-trivial spheres inside the Salvetti complex: through the Nerve Theorem, I was able to identify explicit cellular subcomplexes corresponding to these spheres, thus providing a concrete and visualizable realization of them.