Matings of Cubic polynomials with a fixed critical point, Part I: Thurston Obstructions

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the mating $f \mate g$ contains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.