球面稳定同伦群的研究是同伦论的中心问题之一，也是世界公认的经典难题。以Adams谱序列、Adams－Novikov谱序列为工具发掘出了球面稳定同伦群中的许多非零元素，新元素的发现对于我们更好的了解和研究球面稳定同伦群有不可替代的价值。目前，在研究球面稳定同伦群的工作中，仍然有很多经典的问题等待我们去解决。本课题主要侧重以下四方面：一，在Adams谱序列中，利用希腊字母类元素，得到了许多乘积元素族的收敛性。但这些结果要求下标小于素数p，拟打破这一限制，得到更广范围的希腊字母类乘积元素族在Adams谱序列中的收敛性；二，发掘新的a_0相关元素，以此作为几何输入来寻求球面稳定同伦群新元素族；三，利用Ravenel谱，球谱的Bousfield局部化的同伦群来逼近球面稳定同伦群；四，探讨h_0h_n（n<5）元素在Adams谱序列中的收敛性。

The stable homotopy groups of spheres is one of the central problem in homotopy theory, and it is the well known difficult problem. By using the Adams spectral sequence and the Adams -Novikov spectral sequence, we can detect some new families of nontrivial elements in the stable homotopy groups of spheres, which is important to study the stable homotopy groups of spheres. In the detection and studying of the stable homotopy groups of spheres，there are so many difficult and classical problems which need to be solved. This project specializes in the following four problems: Firstly, in the Adams spectral sequence, we can detect some new products by the help of the Greek letter elements. However, there is a limited condition that the subscript s<p. So we want to cancel this condition and detect general elements of products with Greek letter elements for s>p； Secondly, to detect new a_0 related elements as geometric input, to give new families of nontrivial elements in the stable homotopy groups of spheres；Thirdly，to access the stable homotopy groups of spheres by virtue of the homotopy groups of Ravenel spectrum, sphere spectrum with the Bousfield functor action；Finally, to study the convergence of h_0h_n (n<5) in the Adams spectral sequence.