Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces

TL;DR

This paper studies the homology of moduli spaces of framed, r-Spin, and Pin surfaces, showing they exhibit homological stability and identifying their stable homology with infinite loop spaces, revealing algebraic properties of their mapping class groups.

Contribution

It defines new moduli spaces for these surfaces and proves their homological stability, linking their stable homology to infinite loop spaces and analyzing the algebraic structure of their mapping class groups.

Findings

Stable homology identified with infinite loop spaces.

Stable framed mapping class group has trivial rational homology.

Stable Pin mapping class groups' homology matches non-orientable case.

Abstract

We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spaces each have path components which are Eilenberg--MacLane spaces for the framed, r-Spin and Pin mapping class groups respectively, and hence we also identify the stable group homology of these groups. In particular: the stable framed mapping class group has trivial rational homology, and its abelianisation is Z/24; the rational homology of the stable Pin mapping class groups coincides with that of the non-orientable mapping class group, and their abelianisations are Z/2 for Pin^+ and (Z/2)^3 for Pin^-.