The stable cohomology of the moduli space of curves with level structures

TL;DR

This paper demonstrates that within a stable range, the rational cohomology of moduli spaces of curves with level structures matches that of the classical moduli space, revealing stability properties of their topological invariants.

Contribution

It establishes the stability of rational cohomology for moduli spaces of curves with level structures, extending known results to a broader class of moduli spaces.

Findings

Rational cohomology of moduli spaces with level structures stabilizes in a certain range.

The cohomology of these spaces coincides with that of the classical moduli space.

Stability results hold under specific conditions in the stable range.

Abstract

We prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves.