Ib Madsen, Michael S. Weiss
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group.
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The (Co)homology of the Deligne-Mumford Moduli Spaces of Marked Rational Curves
本文整理了实带标记有理曲线的Deligne-Mumford模空间（上）同调的关键结果与未解决问题，涵盖拓扑与代数两个方向，并尝试用类operad语言刻画其同调结构。
An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.
The Cohomology Ring of the Deligne-Mumford Moduli Space of Real Rational Curves with Conjugate Marked Points
本文证明了具有共轭标记点的实有理曲线Deligne-Mumford模空间的有理上同调环可由复边界除子与实边界超曲面生成，并模去明显几何关系及先前识别的正维数正余维数关系。
Mumford--Morita--Miller classes in generalised cohomology theories
本文研究 Riemann 曲面模空间上的 Mumford--Morita--Miller 类在广义复定向上同调理论中的定义与性质，并强调新现象。
We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.
The Mumford conjecture (after Bianchi)
本文重构并简化呈现 Andrea Bianchi 借助分支覆盖模空间证明 Mumford 猜想的方法。
Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.
Blowdowns of the Deligne-Mumford Spaces of Real Rational Curves
本文构造了实有理曲线Deligne-Mumford模空间的一列光滑商爆破，给出Keel爆破构造的实类比，并显式描述了中间空间及三类爆破，为后续完全确定实模空间的有理（上）同调环奠定基础。
We give a description of the Deligne-Mumford properad expressing it as the result of homotopically trivializing S1 families of annuli (with appropriate compatibility conditions) in the properad of smooth Riemann surfaces with parameterized boundaries. This gives an analog of the results of Drummond-Cole and Oancea--Vaintrob in the setting of properads. We also discuss a variation of this trivialization which gives rise to a new partial compactification of Riemann surfaces relevant to the study of operations on symplectic cohomology.
Inspired by his vanishing results of tautological classes and by Harer's computation of the virtual cohomological dimension of the mapping class group, Looijenga conjectured that the moduli space of smooth Riemann surfaces admits a stratification by affine subsets with a certain number of layers. Similarly, Roth and Vakil extended the conjecture to the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most k rational components. As a consequence of Lefschetz's theorem, Roth-Vakil's conjecture would also imply that the pre…
In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g=3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M_3.
We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has connections with spectral geometry and combinatorics.
Moduli stack of stable curves from a stratified homotopy viewpoint
本文加强了Charney和Lee的结果，证明了稳定曲线范畴能捕获模栈的分层同伦类型，并通过外行进等价对可构造层进行分类。
Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of $\overline M_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n >> 0$. An FS^op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that glue on marked P^1's, we give the homology of $\overline M_{g,n}$ the structure of an FS^op module and bound its degree of generation. As a consequence, we prove that the generating function $\sum_{n} \dim(H_i(\overline M_{g,n})) t^n$ is rational, and its denominator has roots in …
Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford…
We compute the rank of the first homology group and we study the higher Betti numbers of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R)(after Kapranov).
The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.
The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theo…
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