Genus stabilization for the homology of moduli spaces of orbit-framed curves with symmetries-I

TL;DR

This paper extends previous work on the irreducibility of moduli spaces of G-symmetric curves by establishing homology stabilization for a framed variant, advancing understanding of their topological properties.

Contribution

It proves homology stabilization for a framed moduli space of G-symmetric curves, generalizing prior irreducibility results and initiating a broader stabilization program.

Findings

Homology stabilization is established for a G-framed moduli space.

The result parallels Harer's stabilization for classical moduli spaces.

Supports the conjecture of stabilization for all homology groups of these spaces.

Abstract

In a previous paper, arXiv:1301.4409, we showed that the moduli space of curves C with a G-symmetry (that is, with a faithful action of a finite group G), having a fixed generalized homological invariant, is irreducible if the genus g' of the quotient curve C' : = C/G satisfies g'>>0. Interpreting this result as stabilization for the 0-th homology group of the moduli space of curves with G-symmetry, we begin here a program for showing genus stabilization for all the homology groups of these spaces, in similarity to the results of Harer for the moduli space of curves. In this first paper we prove homology stabilization for a variant of the moduli space where one G-orbit is tangentially framed.