SPHERICAL TRIANGLES AND THE TWO COMPONENTS OF THE SO(3)-CHARACTER SPACE OF THE FUNDAMENTAL GROUP OF A CLOSED SURFACE OF GENUS 2

Abstract

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 two 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 two-spheres or two-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.