Topological aspects of the dynamical moduli space of rational maps

A Very Simple Estimate Of Rational Homological Dimension Of Moduli Spaces Of Riemann Surfaces With Boundary And Marked Points

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.

主$G$-丛模栈的有理同伦与Hodge理论

Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles

本文利用拓扑栈的同伦理论，确定了特殊类型复射影簇上主$G$-丛模栈的有理上同调与Hodge-Tate结构。

带标记有理曲线的Deligne-Mumford模空间的（上）同调

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。

具有共轭标记点的实有理曲线Deligne-Mumford模空间的上同调环

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。

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.

曲线的分支覆盖模空间的稳定有理上同调分解

Stable rational cohomology splitting of moduli of branched covers of curves

本文研究了曲线分支覆盖模空间的稳定有理上同调，给出了其关于中间覆盖次数的分解，并用因子化同调及带标签的构型空间加以解释，证明了 d=4,5 时稳定奇维上同调类的存在。

On the rational homotopy type of a moduli space of vector bundles over a curve

We study the rational homotopy of the moduli space ${\mathcal N}_X$ of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface $X$ of genus $g\geq 2$. The symplectic group $Aut(H_1(X,{\mathbb Z}))=Sp(2g,{\mathbb Z})$ has a natural action on the rational homotopy groups $π_n({\mathcal N}_X) \otimes {\mathbb Q}$. We prove that this action extends to an action of $Sp(2g,{\mathbb C})$ on $π_n({\mathcal N}_X) \otimes {\mathbb C}$. We also show that $π_n({\mathcal N}_X) \otimes {\mathbb C}$ is a non-trivial $Sp(2g,{\mathbb C})$-representation fo…

A Moduli Space for Rational Homotopy Types with the Same Homotopy Lie Algebra

Since Quillen proved his famous equivalences of homotopy categories in 1969, much work has been done towards classifying the rational homotopy types of simply connected topological places. The majority of this work has focused on rational homotopy types with the same cohomology algebra. The models in this case were differential graded algebras and acted similarly to differential forms. These models were then used together with some deformation theory to describe a moduli space for all rational homotopy types with a given cohomology algebra. Indeed, this theory has been very well developed. How…

Moduli of abelian covers of elliptic curves

For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M_{1,n}, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geomet…

Aspects of the topology and combinatorics of Higgs bundle moduli spaces

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.