Universal Abelian H-spaces

Level structures on $p$-divisible groups from the Morava $E$-theory of abelian groups

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava $E$-cohomology of finite abelian groups has played an important role in the study of power operations for Morava $E$-theory. The goal of this paper is to explore the relationship between level structures on the $p$-divisible group given by the trivial extension of the universal deformation by a constant $p$-divisible group and the Morava $E$-cohomology of the iterated free loop space of the classifying space of a finite abelian group.

Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics

The moduli stack $\mathcal A_2$ of principally polarized abelian surfaces comes equipped with the universal abelian surface $π: \mathcal X_2 \to \mathcal A_2$. The fiber of $π$ over a point corresponding to an abelian surface $A$ in $\mathcal A_2$ is $A$ itself. We determine the $\ell$-adic cohomology of $\mathcal X_2$ as a Galois representation. Similarly, we consider the bundles $\mathcal X_2^n \to \mathcal A_2$ and $\mathcal X_2^{\operatorname{Sym}(n)} \to \mathcal A_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface $A$ is $A^n$ and $\operatorname{Sym}^…

普适上同调理论

Universal cohomology theories

本文为任意范畴构造普适（上）同调理论，通过函子可表示性取值于动机性$R$-线性阿贝尔范畴，并恢复Grothendieck $\partial$-函子与卫星函子。

A relative theory of universal central extensions

Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. Thus we unify classical, recent and new results in one conceptual framework. The theory we develop is relative to a chosen Birkhoff subcategory of the category considered: for instance, we consider groups vs. abelian groups, Lie algebras vs. vector spaces, precrossed modules vs. crossed modules and Leibniz algebras vs. Lie algebras. We consider a fundamental condition on composition of central extensions and give examples of categories …

万有同伦理论与相关同调代数

Universal Homotopy Theories and Associated Homological Algebras

本文针对小范畴，将其万有同伦范畴与含幺交换环上的线性阿贝尔范畴相关联，从而在该范畴中开展相应的同调代数研究。

一个一般的万有系数定理及其应用

A general Universal Coefficient Theorem, and applications

本文在阿贝尔范畴中得到一般万有系数定理，应用于阿贝尔波兰群范畴左心，获得相关短正合序列，用于刻画Čech上同调幻子群、描述其与高阶幻映射关系及对螺线管补高阶版本上的映射进行同伦分类并衡量分类问题复杂性。