On the topology of moduli spaces of real algebraic curves

TL;DR

This paper proves that mapping class groups of all real algebraic curves are virtual duality groups and introduces a new complex to analyze their topology, revealing the homotopy type of associated spaces.

Contribution

It establishes the virtual duality property for mapping class groups of real algebraic curves and defines the $\\mathcal{A}\\mathcal{B}\\mathcal{C}$-complex to study their topological features.

Findings

Mapping class groups are virtual duality groups.

The $\\mathcal{A}\\mathcal{B}\\mathcal{C}$-complex is homotopy equivalent to an infinite wedge of spheres.

The dimension of the complex depends on the surface's topological type.

Abstract

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surface with boundary called the $\mathcal{A}\mathcal{B}\mathcal{C}$-complex. This complex encodes the intersection patterns of isotopy classes of essential simple arcs, boundary components, and essential simple closed curves. We show that this complex is homotopy equivalent to an infinite wedge of spheres; its dimension is dependent on the topological type of the surface.