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

Abstract

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 − .

Resolutions of moduli spaces and homological stability

“…In a companion paper [22] we verify the hypotheses of Theorems 7.1 and 7.2 for the tangential structures given by framings, Spin structures (and more generally r-Spin structures), and Pin ± structures. We then give computational applications of these stability results, for example: the framed mapping class group has trivial stable rational homology, and its stable abelianisation is Z/24; the Pin + mapping class group has stable abelianisation Z/2, and the Pin − mapping class group has stable abelianisation (Z/2) 3 .…”

On Positive Braids, Monodromy Groups and Framings

“…We will now briefly recall the basics of the theory of framed surfaces, concentrating in particular on the action of the mapping class group on such structures, as investigated in [6] and [21]. In what follows, we will adhere to the notations and conventions of [6], but we will restrict only to the case of surfaces with connected boundary.…”

Relative homological representations of framed mapping class groups

“…The basic theory of relative isotopy classes of framings was established by Randal–Williams [11]. To state his results, we define a distinguished geometric basis on to be a collection of oriented simple closed curves and legal arcs , subject to the following conditions.…”

“…The following is a summary of the basic theory of relative winding number functions and relative isotopy classes of framings. For further discussion, see [3, Section 2] or [11]. Proposition Fix and , and let be a framing of .…”