Mod p homology of the stable mapping class group

A twisted first homology group of the handlebody mapping class group

Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for $g\geq 2$.

Embeddings of braid groups into mapping class groups and their homology

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

Homology of the mapping class group for surfaces of genus 2 with boundary

We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture.

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 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…

Mapping spaces and homology isomorphisms

Let Map(K,X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E_* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X) will send a E_*--isomorphism in either variable to a map that is monic in E_* homology. Interesting examples arise by letting E_* be K--theory, K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces.

Homological stability for quotients of mapping class groups of surfaces by the Johnson subgroups

We study quotients of mapping class groups (Γ_{g,1}) of oriented surfaces with one boundary component by terms of their Johnson filtrations, and we show that the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity. We also compute the stable (co)homology with constant rational coefficients for one family of such quotients.

Homology stability for symmetric diffeomorphism and mapping class groups

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of $W$ connected sum with $k$ copies of an arbitrary compact smooth manifold $Q$ of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

Sobolev mappings, degree, homotopy classes and rational homology spheres

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$, $n=\dim M$. In particular, we prove that if $M$ and $N$ are compact oriented manifolds without boundary and $\dim M=\dim N=n$, then the degree is well defined in $W^{1,P}(M,N)$ if and only if the universal cover of $N$ is not a rational homology sphere, and in the case $n=4$, if and only if $N$ is not homeomorphic to $S^4$.

$\mathbb{Z}/p$-同调3-球面源自Level-p映射类群阿贝尔化的不变量

Invariants of $\mathbb{Z}/p$-Homology 3-Spheres from the Abelianization of the Level-p Mapping Class Group

本文研究了定向$\mathbb{Z}/d$-同调3-球面与level-$d$映射类群的关系，提出了一种基于平凡2-上闭链族构造不变量的工具，并借此求出了素数$p$下由level-$p$映射类群阿贝尔化构造的所有不变量，推翻了B. Perron的相关猜想。

The rational stable homology of mapping class groups of universal nil-manifolds

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.

旋映射类群和二次辛群的同调稳定性

Homological stability of spin mapping class groups and quadratic symplectic groups

本文通过细胞E₂-代数研究了旋映射类群和二次辛群的同调稳定性，改进了稳定性结果并给出了第一同调群的完整描述。

Lagrangian mapping class groups from a group homological point of view

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product …

Abelian quotients of mapping class groups of highly connected manifolds

We compute the abelianisations of the mapping class groups of the manifolds $W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping cylinders and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable $J$-homomorphism. We relat…

单位切丛同调中映射类群的稳定扭曲上同调

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。

Surfaces in a background space and the homology of mapping class groups

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, γ). This is the space consisting of triples, (F_{g,n}, ϕ, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, ϕis a parameterization of the boundary, and f : F_{g,n} \to X is a continuous map that satisfies a boundary condition γ. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S_{\infty, n}(X; γ), defined to be the limit as the genus g gets large, of the spaces S_{g,n} (X; γ). Our re…

论大映射类群的同调

On the homology of big mapping class groups

本文证明单孔康托树曲面的映射类群是无环的，并由此确定单穿孔康托树曲面映射类群的同调，回答了Calegari和Chen的问题，同时将结果推广到一般二叉树曲面。

The homology of the Milnor fiber for classical braid groups

In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.

Improved homological stability for the mapping class group with integral or twisted coefficients

In this paper we prove stability results for the homology of the mapping class group of a surface. We get a stability range that is near optimal, and extend the result to twisted coefficients.

大型映射类群同调中的紧致与有限型支撑

Compact and finite-type support in the homology of big mapping class groups

本文研究了无限型曲面映射类群同调中是否存在支撑在紧致或有限型子曲面上的非平凡类，利用可移动子曲面和同调稳定性给出了亏格为正时的几乎完整解答及亏格为零时的部分解答。

Homological stability of non-orientable mapping class groups with marked points

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.

Abelian quotients of monoids of homology cylinders

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the f…

Geometric Homology

The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately addressed. In particular, our goal is to construct a chain level theory based on smooth mappings of manifolds with corners. In addition, we will avoid using the fact that smooth manifolds with can be triangulated. As we shall see, transversality and intersections play a major role in setting up this theory. From a pedagogical viewpoint, having intersection theory …

On truncated mapping spaces of $E_n$ operads

The mapping spaces of the $r$-truncated versions of the $E_n$ operads appear as the $r$-th stage of the Taylor tower for long embedding spaces. It has been shown that their rational homotopy groups can be expressed through graph homology in the limit $r\to \infty$. For finite $r$ only a part of the graph homology appears in the homotopy groups of the Taylor tower at this stage, possibly along with some additional unstable homotopy groups. In this paper we study the convergence properties in some more detail. In particular, we provide bounds for the stage $r$ at which the various graph homology…

具有不可数整同调的大映射类群

Big mapping class groups with uncountable integral homology

本文证明无限型曲面的紧支撑映射类群闭包和 Torelli 群的整同调在每个正次数上不可数，并将该结论推广到末端空间含拓扑可区分点极限点的一大类曲面的完整映射类群，同时构造了亏格2曲面纯映射类群第一同调中的10阶元素。

An infinite genus mapping class group and stable cohomology

We exhibit a finitely generated group $\M$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface $\su$ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus $g$ with $n$ boundary components, for any $g\geq 0$ and $n>0$. We construct a representation of $\M$ into the restricted symplectic group ${\rm Sp_{res}}({\cal H}_r)$ of the real Hilbert space generated by the homology classes of non-separating circles on $\su$, which generalizes the classical …

曲面构型上的海森堡同调

Heisenberg homology on surface configurations

受经典辫群表示启发，本文研究带边界可定向曲面上无序构型的同调，利用离散海森堡群表示定义非交换局部系统，构造了扭曲与非扭曲的映射类群表示，并证明其保持半双线性形式。

Homological stability for the mapping class groups of non-orientable surfaces

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 2-torsion in the second homology of the genus $3$ mapping class group

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…

Configuration-mapping spaces and homology stability

For a given bundle $ξ\colon E \to M$ over a manifold, configuration-section spaces on $ξ$ parametrise finite subsets $z \subseteq M$ equipped with a section of $ξ$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the $2$-disc with specified deck transformation group. We prove that …

$E_2$-cells and mapping class groups

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.

A geometric approach to equivariant factorization homology and nonabelian Poincaré duality

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $E_V$-algebra using a two-sided bar construction, generalizing [And10, KM18]. This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincare dual…

Cohomological structure of the mapping class group and beyond

In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures…

开-闭 Hochschild 同调与相对圆盘映射空间

Open-Closed Hochschild Homology and the Relative Disk Mapping Space

本文建立 Chen 自由环路空间迭代积分模型的相对版本，利用开-闭同伦代数与开-闭 Hochschild 同调为相对圆盘映射空间构造模型，并在特定条件下证明其为拟同构，推广了 Chen 及 Getzler-Jones 的经典结果。

The second rational homology group of the moduli space of curves with level structures

Let $Γ$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Γ$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Γ;\Q) \cong \Q$ for $g \geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\Q$. We also prove analogous results for surface with punctures and boundary components.

曲面的阿贝尔覆盖与 Torelli 群的同调

Abelian covers of surfaces and the homology of the Torelli group

本文研究了映射类群和 Torelli 群在曲面泛阿贝尔覆盖有理同调系数下的第一同调群，证明了单边界与单穿孔情形下维度的有限与无限对比性，为计算 Torelli 群第二有理同调群提供关键支撑。

Equivariant group presentations and the second homology group of the Torelli group

We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,\mathbb{Z})$-module.

The mapping class group of manifolds which are like projective planes

In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.

On the singular homology of one class of simply-connected cell-like spaces

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {\sl Snake space}. In the sequel we introduced the functor $SC(-,-)$ defined on the category of all spaces with base points and continuous mappings. For the circle $S^1$, the space $SC(S^1, \ast)$ is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces $SC(Z, \ast)$ for any path-connected compact spaces $Z$.

Central stability for the homology of congruence subgroups and the second homology of Torelli groups

We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.

Universal Realisators for Homology Classes

We study oriented closed manifolds M^n possessing the following Universal Realisation of Cycles (URC) Property: For each topological space X and each integral homology class z of it, there exist a finite-sheeted covering \hM^n of M^n and a continuous mapping f of \hM^n to X such that f takes the fundamental class [\hM^n] to kz for a non-zero integer k. We find wide class of examples of such manifolds M^n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we prove that for each 4-dimensional …

单位切丛同调外幂中映射类群的稳定扭曲上同调

Stable twisted cohomology of the mapping class groups in the exterior powers of the unit tangent bundle homology

本文研究映射类群在单位切丛有理同调外幂系数下的稳定扭曲上同调，显式计算 d≤5 的 Tor 群，并证明除 d=2 外相应模均非自由。

Non-trivial action of the Johnson filtration on the homology of configuration spaces

We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.

Tethers and homology stability for surfaces

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…

Braids, mapping class groups, and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with any trivial coefficients. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping. In an appendix we discuss geometrically defined homomorphisms from the braid to the mapping class group.

Topological K-(co-)homology of classifying spaces of discrete groups

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculat…

等变椭圆上同调与映射叠

Equivariant Elliptic Cohomology and Mapping Stacks

本文引入一种新的叠上同调理论——椭圆Hochschild同调，证明其基本性质并计算若干例子，进而引入周期循环版本，证明在复数域上对商叠恢复Grojnowski的解析化等变椭圆上同调。

Mapping class groups of highly connected $(4k+2)$-manifolds

We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.

同调稳定性：一种计算工具

Homological stability: a tool for computations

本文综述了同调稳定性定理的证明工具与稳定同调的计算技术，并以 Higman-Thompson 群的同调计算为例展示其应用。

Torelli与同余子群同调中稳定范围的改进

Improved stability ranges in the homology of Torelli and congruence subgroups

通过表示论方法，改进了Torelli子群和同余子群在特定线性群作用下的第二同调群的中心稳定范围。

Homological stability for mapping class groups of surfaces

We give a complete and detailed proof of Harer's stability theorem for the homology of mapping class groups of surfaces, with the best stability range presently known. This theorem and its proof have seen several improvements since Harer's original proof in the mid-80's, and our purpose here is to assemble these many additions.

Morita's trace maps on the group of homology cobordisms

Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization…

形式群与椭圆 Hochschild 同调的 Fourier--Mukai 等价

Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology

本文比较两种椭圆 Hochschild 同调定义，提出形式群的新 Fourier--Mukai 对偶性以统一扭曲 Hochschild 同调，并引入全局版本暗示了普遍的 TMF-Hochschild 同调形式。

Stability in the homology of congruence subgroups

The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena representation stability. We prove that the homology groups of congruence subgroups of $GL_n(R)$ (for almost any reasonabl…

Hairy graphs and the unstable homology of Mod(g,s), Out(F_n) and Aut(F_n)

We study a family of Lie algebras {hO} which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map which generalizes the trace map defined by S. Morita. For the Lie operad we use the trace map to find la…