Elizabeth Hanbury
Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short note we analyse the situation where the underlying non-orientable surfaces have marked points.
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We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i--this is the non-oriented analogue of the Mumford conj…
The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts of open-closed string theory, we give an estimate of rational homological dimension of Riemann suraces with possible boundary and marked points(can lie on both interior and boundary). We hope it will have applications in open-closed theory, for example, open-closed Gromov-Witten theory in the future.
Homological stability for the space of hypersurfaces with marked points
本文研究线性系统中带标记点光滑超曲面空间，通过同调 h-原理与射流丛截面对比，并用有理模型计算稳定范围内的有理上同调及 Hodge 权，建立同调稳定性及其与 Howe 动机结果的联系。
$C$-triviality of manifolds of low dimensions
本文研究了低维闭光滑流形的C-平凡性，通过谱序列和微分的阶数进行同调分类，7维时给出可定向和不可定向流形的条件。
Farrell cohomology of the pure mapping class group of non-orientable surfaces
本文针对带标记点的非可定向曲面纯映射类群，通过分类 p 阶子群共轭类并推广曲面核满同态的拓扑等价概念，确定了奇素数 p 情形下亏格为 p 的 Farrell 上同调的 p-准素分量。
This paper is a survey of the relationship between labelled configuration spaces, mapping class groups with marked points and function spaces. In particular, we collect calculations of the cohomology groups for the mapping class groups of low-genus orientable and non-orientable surfaces.
We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. Also, it is well known that the mapping class group $\text{Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{Mod} (S_{g-1}; 2k)$, the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of $\text{Mod}(N_g; k)$ can be lifted isomorphically to a subgro…
Twisted homological stability for handlebody mapping class groups
该研究证明了手柄映射类群的扭转同调稳定性，并通过引入系数双系统，扩展了手柄稳定性结果，应用于三维手柄模空间的同调稳定性。
We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations. In the first part, we introduce and study plant complexes, a large new class of simplicial complexes, generalizing the arc complex on a surface with marked points. In the second part, we generalize a result by Ellenberg-Venkatesh-Westerland by showing that homological stabilization…
We prove a new kind of stabilisation result, "secondary homological stability", for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_2$-algebras, which have no $E_2$-cells below a certain vanishing line.
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B_n(M) satisfies classical homological stability: for each i, H_i(B_n(M); Q) is isomorphic to H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural exa…
We show that the pure mapping class group $\mathcal{N}_{g}^{k}$ of a non-orientable closed surface of genus $g\geqslant 2$ with $k\geqslant 1$ marked points has $p$-periodic cohomology for each odd prime $p$ for which $\mathcal{N}_{g}^{k}$ has $p$-torsion. Using the Yagita invariant and the cohomology classes obtained by the representation of subgroups of order $p$, we obtain that the $p$-period is less than or equal to $4$ when $g\geqslant 3$ and $k\geqslant 1$. Moreover, combining the Nielsen realization theorem and a characterization of the $p$-period given in terms of normalizers and centr…
Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class …
Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set.
In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of non-zero $d$-dimensional homology corresponds exactly to the presence of a $d$-dimensional cycle in the simplicial complex. We also show that $d$-d…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration spaces are moduli spaces of manifolds with marked points, and in [Tillmann, 2016] this relation was used to show that the homology of these moduli spaces is also stable as the number of marked points goes to infinity. Since a disc neighbourhood of a marked point may be viewed as the cone on its boundary sphere, we may think of marked points as inessential conic…
In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $S_g (K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $S_{g,n} (K)$ was studied by the authors in \cite{cohenmadsen}. The study there relied on stability results for the homology of mapping class groups, $Γ_{g,n}$ with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when $n$, the number of boundary components, i…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.
The action of the Mapping Class Group on the fundamental group of the complement of a finite subset of a Riemann surface of positive genus
本文通过将映射类群 $M(g,n)$ 生成元的像计算为基本群的外自同构，描述了该群在具有 $n$ 个标记点的正亏格紧致可定向拓扑曲面 $T_{g,n}$ 基本群上的作用。
Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $χ(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With additional assumptions to ensure orientability of some components of the singular set (e.g. if $G$ acts by symplectic symmetries, or preserving a spin structure), we also rule out $C_2 \times C_2$ actions. The proofs use equivariant cohomology, localization, and a careful study of the fi…
Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology
本研究计算了带边紧致可定向曲面映射类群以单位切丛同调为系数的稳定扭曲上同调，发现其为协变函子超出传统框架，还计算了符合传统框架的一阶上同调系数的稳定上同调群。
Homological stability for sequences of groups is often proved by studying the spectral sequence associated to the action of a typical group in the sequence on a highly-connected simplicial complex whose stabilizers are related to previous groups in the sequence. In the case of mapping class groups of manifolds, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. In this paper we focus on the case of surfaces and show that by using more refined geometric objects consisting of certain configurations of curves with arcs that tether these c…
Stability in the homology of configuration spaces of high dimensional manifolds
该研究针对高维流形配置空间的同调稳定性问题，拓展了已有结论，证明了上同调集中在偶度数的流形其配置空间奇同调稳定域与流形维数相关，且二阶上同调非平凡时偶同调稳定域最优且仅由配置点数决定。
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 homo…
Torsion at the Threshold for Mapping Class Groups
本文研究带标记点的闭可定向曲面映射类群的离散泛欧拉类幂次$n=g$的情形，证明该幂次对应的类为挠类，其生成的循环子群的阶是$4g(2g+1)(2g-1)$的正整数倍。
Let M_g^n be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g \geq 2, the cohomology groups {H^i(M_g^n;Q)}_{n=1}^{\infty} form a sequence of Sn representations which is representation stable in the sense of Church-Farb [CF]. In particular this result applied to the trivial Sn representation implies rational "puncture homological stability" for the mapping class group Mod_g^n. We obtain representation stability for sequences {H^i(PMod^n(M);Q)}_{n=1}^{\infty}, where PMod^n(M) is the mapping class group of many connected manifolds M of dimension d…
This work is NOT to be used as reference. First, because as C.F.~Bödigheimer and M.~Korkmaz pointed to us the computation of the $\mathbf{Z}_2$ factor that remained undecided in M.~Korkmaz and A. Stipsicz, {\em The second homology groups of mapping class groups of orientable surfaces.} Math. Proc. Camb. Phil. Soc., was shown to exist by Skasai, see hi Theorem 4.9 and Corollary 4.10 in {\em Lagrangian mapping class groups from a group homological point of view.} Algebr. Geom. Topol. 12 (2012), no. 1, 267--291. Second, because one could obtain this result by gathering old results in the literatu…
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