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

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 that 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 abelianization 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 abelianizations are ℤ/2 for Pin + and (ℤ/2)3 for Pin −.