Homology of the moduli spaces and mapping class groups of framed, \(r\)-spin and pin surfaces (Q2879007)

Moduli spaces and mapping class groups of oriented surfaces have been studied intensively in recent times, as well as moduli spaces and mapping class groups of surfaces equipped with other tangential structures, including unoriented surfaces and Spin surfaces. Here, a \textit{tangential structure} means a Serre fibration \(\theta: B\to BO(2)\), which in the above cases is induced by a group homomorphism to the topological group \(O(2)\). As the title suggests, this paper studies the moduli spaces and mapping class groups for certain other tangential structures.NEWLINENEWLINEFor a closed surface \(F\), the moduli space \(\mathcal{M}^\theta(F)\) of \(\theta\) structures on \(F\) is the homotopy quotient by \(\mathrm{Diff}(F)\) of the space of bundle maps from \(TF\) to \(\theta^*\gamma_2\), the pullback of the universal bundle \(\gamma_2\) over \(BO(2)\). For a compact surface with boundary the definition is the same, except the bundle maps are prescribed on \(\partial F\), and the moduli space is written \(\mathcal{M}^\theta(F;\delta)\) for a boundary condition \(\delta\). The \(\theta\) considered in this paper are as follows. For \(r\geqslant 0\) the map \((-)^r : U(1) \to U(1) = SO(2) \hookrightarrow O(2)\) induces a Serre fibration over \(BO(2)\) called \(\theta^r\). When \(r=0\) this is the structure of a framing, when \(r=1\) it is an orientation and when \(r=2\) it is a (usual) Spin structure. When \(r=2\) the double covering is usually denoted \(\mathrm{Spin}(2) \to SO(2)\), and can be extended in two different ways to \(O(2)\), written \(\mathrm{Pin}^+(2)\to O(2)\) and \(\mathrm{Pin}^-(2)\to O(2)\). These induce Serre fibrations over \(BO(2)\) called \(\theta=\mathrm{Pin}^\pm\).NEWLINENEWLINEThe main result is that the moduli spaces of \(\theta^r\) and \(\mathrm{Pin}^\pm\) structures are homologically stable, with respect to the genus of a surface, in explicitly given ranges. The \(\theta^r\) structures only apply to orientable surfaces, and the results for \(\mathrm{Pin}^\pm\) structures are only stated for non-orientable surfaces, since they reduce to \(\theta^2\) structures in the orientable case. Given a \(\theta\) structure \(\xi\) on a surface \(F\), its mapping class group is the fundamental group of \(\mathcal{M}^\theta(F;\xi|_{\partial F})\) based at \([\xi]\). Since each moduli space under consideration is a disjoint union of aspherical spaces, the corresponding mapping class groups are also homologically stable.NEWLINENEWLINEIn each case, the methods of [\textit{S. Galatius} et al., Acta Math. 202, No. 2, 195--239 (2009; Zbl 1221.57039)] (or alternatively [\textit{S. Galatius} and \textit{O. Randal-Williams}, Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)] in the \(\theta^2\) and \(\mathrm{Pin}^\pm\) cases) give as a corollary that the homology in the stable range is the homology of certain path-components of the infinite loopspace of the spectrum \(\mathrm{MT}\theta\), the Thom spectrum of the virtual bundle \(-\theta^*\gamma_2\). The abelianisation of the mapping class groups, in the stable range, is therefore \(\pi_1(\mathrm{MT}\theta)\). In the framed case the spectrum \(\mathrm{MT}\theta^0\) is \(\mathrm{S}^{-2}\), whose \(\pi_1\) is well-known to be \(\mathbb{Z}/24\). In the general \(\theta=\theta^r\) case the abelianisation in the stable range was calculated in [\textit{O. Randal-Williams}, Adv. Math. 231, No. 1, 482--515 (2012; Zbl 1280.55011)] and depends on the value of \(r\) modulo \(12\). In the \(\mathrm{Pin}^\pm\) cases it is calculated in the appendix of the paper under review using the Adams spectral sequence: it turns out to be \(\mathbb{Z}/2\) for \(\theta=\mathrm{Pin}^+\) and \((\mathbb{Z}/2)^3\) for \(\theta=\mathrm{Pin}^-\). Some more computations of the stable homology in the \(\mathrm{Pin}^\pm\) cases are also given.NEWLINENEWLINEThe strategy to prove homological stability is to apply the general method set up in [\textit{O.\ Randal-Williams}, ``Resolutions of moduli spaces and homological stability'', \url{arXiv:0909.4278}]. The necessary inputs to apply this method are to show, for the tangential structures \(\theta\) under consideration, that the moduli spaces \(\mathcal{M}^\theta\) are ``\(k\)-trivial'' for some \(k\) and that they stabilise on \(\pi_0\). In fact, the \(k\)-triviality condition is formally implied by stabilisation on \(\pi_0\), so this is the essential input, which the machine of [op.\ cit.] will bootstrap to full homological stability. Thus, most of the paper is concerned with carefully proving \(\pi_0\)-stability for each \(\theta\) under consideration, meaning that the functions between the sets \(\pi_0(\mathcal{M}^\theta(F,\delta))\) induced by gluing \(\theta\)-surfaces are bijections once the genus is sufficiently large. These sets can be viewed as the set \(\theta(F,\delta)\) of isotopy classes of \(\theta\) structures on the surface \(F\), quotiented out by the action of the (unoriented) mapping class group \(\Gamma(F)\) of \(F\). In the framed and \(r\)-Spin cases the author views it alternatively as the set of isotopy classes of \(\theta^r\) structures extending a fixed orientation of the surface (so in this case \(\theta^r(F,\delta)\) denotes this set instead), quotiented out by the oriented mapping class group \(\Gamma^+(F)\).NEWLINENEWLINEHere we mainly describe the case \(\theta = \theta^r\), and write \(\Sigma_{g,b}\) for an oriented surface of genus \(g\) with \(b\) boundary components. The proof in the \(\mathrm{Pin}^\pm\) cases follows the same structure. There are three steps: (1) To construct a bijection between \(\theta^r(\Sigma_{g,b},\delta)\) and \((\mathbb{Z}/r)^{2g+\mathrm{min}(b-1,0)}\), to be thought of as ``coordinates'' on the set of isotopy classes of \(\theta^r\) structures. (2) To compute the action of the mapping class group via these coordinates, and use this to calculate the orbit set (which is \(\pi_0\) of the moduli space) for sufficiently large genus. This turns out to be either trivial or \(\mathbb{Z}/2\) in each case (except \(\mathrm{Pin}^-\) for which it is \(\mathbb{Z}/4\)), and the orbits are distinguished by an explicitly-defined invariant of the \(\theta\)-surface (the \textit{generalised Arf invariant}). (3) To study the effect on this invariant of gluing surfaces together, and deduce that it induces bijections on \(\pi_0\) of the moduli spaces (for sufficiently large genus).NEWLINENEWLINEMore details on step (1): The ``coordinates'' of a given \(\theta\) structure are defined by taking its restriction to a particular system of directed arcs and loops on the surface (starting at a basepoint on the boundary), which is then determined by the monodromies along these arcs and loops (using a prearranged trivialisation at the endpoints of the arcs). The coordinates give a bijection because any \(\theta\) structure on the system of arcs and loops extends uniquely (up to isotopy) to one on the whole surface (since cutting along the arcs and loops reduces the surface to a disc, and any \(\theta\) structure on the boundary of a disc can be extended uniquely up to isotopy to its interior).NEWLINENEWLINEMore details on step (2): For genus at least \(2\) (and also genus \(1\) when \(r=2\)) the orbit set (i.e., set of path-components of the moduli space) is \(\mathbb{Z}/2\) for \(r\) even (including the framed case \(r=0\)) and a single point for \(r\) odd. The invariant \(A\) distinguishing the two orbits is called the \textit{generalised Arf invariant}, since it is equal to the Arf invariant when \(r=2\) (ordinary Spin structures) and the surface has zero or one boundary components. For \(\mathrm{Pin}^\pm\), once the (non-orientable) genus is at least \(3\), the orbit set is \(\mathbb{Z}/2\) for \(\mathrm{Pin}^+\) and is \(\mathbb{Z}/4\) for \(\mathrm{Pin}^-\).NEWLINENEWLINESince the technical results needed for the homological stability proof are that certain maps between sets of size \(2\) (or \(4\)) are bijections, one has to be very careful not to make non-well-defined identifications of these sets with \(\mathbb{Z}/2\) (or \(\mathbb{Z}/4\)) -- hence the very explicit description of the invariant \(A\) in the paper, which gives such identifications. The author corrects an error in [\textit{L. Dabrowski} and \textit{R. Percacci}, J. Math. Phys. 29, No. 3, 580--593 (1988; Zbl 0666.58010)] related to computing the stable \(\pi_0\) of the moduli space of \(\mathrm{Pin}^+\) structures, which stems from using a bijection with \(\mathbb{Z}/2\) at some point in their proof, which turns out not to be well-defined.