Master's Thesis

The Sullivan Conjecture posits the existence of complete intersections that, while distinct as complex manifolds, share isomorphic underlying smooth structures. In order to study these complete intersections, we rely on an associated normal map, which is the data maps of the form $\gamma \to \gamma^{\otimes d}$, where $\gamma$ is a line bundle. We define the split polynomials, a monoid under composition formed by polynomials of the form $z \mapsto z^d$ on a one-dimensional subspace of a complex vector space. A split polynomial serves as a basic model for the behaviour observed on each fibre of the normal map. We then explore the structure of this split polynomial space and its quotient under the unitary group action, which we denote as the $\mathcal A$-space. While not the primary objective, we hope that we may gather evidence supporting the veracity of the Sullivan Conjecture through the study of these spaces.