The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU(2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of 2-spheres or 2-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL(3,R)-representations.