项目研究内容主要有：一、球面稳定同伦群的深入研究；二、有关广义Sullivan猜想问题的研究。球面稳定同伦群是稳定同伦论的中心问题之一，它对代数拓扑本身及其他许多数学分支都有着重要作用。广义Sullivan猜想是有理同伦论中的一个重要问题，它具有很高的理论研究价值。本项目将利用Adams谱序列、May谱序列以及对由模 $p$ Steenrod 代数$A$的所有循环缩减幂$p^i$ ($i>0$)生成的子代数$P$的上同调进行次数估计等工具对球面稳定同伦群进行进一步研究，证明球面稳定同伦群中若干低滤子的不可分解的同伦元素族的存在性和若干高滤子的同伦元素族的存在性。在有关广义Sullivan猜想的研究中，我们主要考虑其在有理情形且$G$为Lie群时的情况。利用空间的Sullivan模型以及空间的李无穷代数模型等知识对映射$k$的代数模型以及同伦不动点集的有理同伦型等进行研究。

In this project, the research content mainly includes the following two parts: 一. further studies on the stable homotopy groups of spheres; 二. some important studies on the generalized Sullivan conjecture. The stable homotopy groups of spheres is one of the central problems in stable homotopy theory, and has very important effects on algebraic topology itself and other branches of mathematics. The generalized Sullivan conjecture is an important problem in rational homotopy theory, and has great theoretical value. In this project, we will detect some indecomposable families of homotopy elements of low filtration in the stable homotopy groups of spheres and some new families of homotopy elements of spheres of high filtration. In order to accomplish these things, we will make use of the Adams spectral sequence, the May spectral sequence and the estimate of the cohomology of $P$, etc., where $P$ denotes the Hopf subalgebra of the mod $p$ Steenrod algebra $A$ generated by the reduced power operations $p^i$ ($ i > 0$). In the studies on the generalized Sullivan conjecture, we focus on the case when $p=0$ and $G$ is a Lie group. We will use the following knowledge, such as the Sullivan model of a space, the $L_{\infty}$ model of a space and so on. In this project we will determine the algebraic models of the map $k$ and the rational homotopy types of the homotopy fixed point sets of some Lie group action, etc.