Lectures on Klein surfaces and their fundamental group

TL;DR

This paper introduces the fundamental group of Klein surfaces, exploring their topological classification, relation to real algebraic curves, and the variety of unitary representations, connecting algebraic and topological perspectives.

Contribution

It provides a comprehensive introduction to the fundamental group of Klein surfaces and links it to representation varieties, advancing understanding in real algebraic and topological geometry.

Findings

Classification of Klein surfaces reviewed

Fundamental group properties established

Connection between unitary representations and Riemann surface groups

Abstract

The goal of these lectures is to give an introduction to the study of the fundamental group of a Klein surface. We start by reviewing the topological classification of Klein surfaces and by explaining the relation with real algebraic curves. Then we introduce the fundamental group of a Klein surface and present its main basic properties. Finally, we study the variety of unitary representations of this group and relate it to the representation variety of the topological fundamental group of the underlying Riemann surface.