球面稳定同伦群的计算是同伦论的一个中心问题，许多数学的不变量都与之有着密切的联系，其计算的主要工具是Adams谱序列与Adams-Novikov谱序列。虽然利用这两类谱序列我们已经得到了球面稳定同伦群的很多非平凡元素，但是迄今为止我们离问题的彻底解决还相去甚远。鉴于此，本项目将在已有研究的基础之上主要研究如下两个问题：(1)计算Adams谱序列E2-项中虑子为4的生成元，并计算这些生成元的二阶非平凡Adams微分。(2)利用代数Novikov谱序列计算beta族元素在Adams谱序列E2-项中的表示，并证明这些beta族元素与其它元素的乘积的收敛性。本项目的优势之处在于能够从代数的角度来理解球面稳定同伦群，以期从不同的角度来寻找解决问题的新思路。

The computation of the stable homotopy groups of spheres is one of the central problems in homotopy theory. Many other invariants in mathematics are closely connected with the stable homotopy groups of spheres which are calculated mainly by Adams spectral sequence and Adams-Novikov spectral sequence. Although we have obtained many nontrivial elements of the stable homotopy groups of spheres via these two kinds of spectral sequences, up to now the thorough solution of this problem is still far beyond approach. In view of this, our project will mainly study the following two problems based on the already known research: (1)Calculate the generators of the E2 –term with filtration four of the Adams spectral sequence as well as the nontrivial secondary Adams differentials of these generators. (2) Calculate the represents of the beta-family in the Adams spectral sequence via the algebraic Novikov spectral sequence. Then we go on to verify the convergence of the product of these beta-family with other elements. The advantage of this project is to understand the stable homotopy groups of spheres from the view of algebra in the hope of finding new method for solution of our problems from different point of view.