Connected components of representation spaces of non-orientable surfaces

TL;DR

This paper proves that the representation space of a non-orientable surface's fundamental group into PSL(2,R) has exactly two connected components distinguished by a Stiefel-Whitney class.

Contribution

It establishes the number of connected components of the representation space for non-orientable surfaces and characterizes them via a specific Stiefel-Whitney class.

Findings

Exactly two connected components in the representation space.

Components correspond to preimages of a Stiefel-Whitney class.

Method extends the Euler class approach to non-orientable surfaces.

Abstract

Let M be a compact closed non-orientable surface. We show that the space of representations of the fundamental group of M into PSL(2,R) has exactly two connected components. These two components are the preimages of a certain Stiefel-Whitney characteristic class, computed in a similar way as the Euler class in the orientable case.