本文对李群的自映射进行研究，通过上同调函子把李群上的自映射对应到诱导的上同 调环自同态，然后再通过 Steenrod 运算来确定哪些李群上同调环自同态不能由连续映射实 现，这给出了李群上同调环自同态的限制。作为例子，主要对???? ?? 、???? ?? 进行讨论，通 过计算及证明得到了在李群???? ?? 、???? ?? 上同调环自同态上的限制。在此基础上，得出了 李群自映射的 Lefschetz 数在该限制下对应的结果。

 第二章主要关注李群的自映射同伦集与李群上同调环自同态集合之间的联系。我们将
二者用上同调函子诱导的同态进行联系，说明了研究李群上同调自同态对研究李群的自映
射有直接帮助，为接下来研究李群上同调自同态上的限制提供必要的理论依据。 

第三章针对???? ?? 进行研究，通过 Steenrod 运算来得到??? ???? ?? 自同态的限制，并且 围绕这一限制，对其上的 Lefschetz 数的限制也做了相应的说明。

第四章针对???? ?? 进行研究，结果与???? ?? 类似。

第五章旨在利用李群的模 ??分解进行讨论。主要介绍了李群的模 ??分解结果，并对如 何利用李群的模 ??分解进行计算做了简单说明。

In this paper, We study the self-mapping of Lie groups. Through the cohomology functor, we correspond the self-mapping of Lie groups to the induced endomorphism of cohomology ring. And then Steenrod operation is used to determine which cohomomorphic ring endomorphisms of Lie groups cannot be realized by continuous mapping. This gives the limit of endomorphism of homomorphic rings of Lie groups. For example, the self-mapping of ???? ?? and ???? ?? is mainly discussed, and we obtained the conclusions through calculation and proof. Besides, relevant calculations, such as Lefshitz number, are also studied and the corresponding results are obtained.

The second chapter mainly focuses on the relation between the homotopy set of lie group and the set of homomorphisms of the cohomology ring of lie group. We connect them by the homomorphism induced by the cohomological functor. And which lay the necessary theoretical basis for the seried study.

In the third chapter, we study ???? ?? , Steenrod operation is used to calculate the limit of the self-map of ??? ???? ?? , and this conclusion is also used to state the limit of Lefschetz number.

In the forth chapter, we study ???? ?? , the result is similar to ???? ?? .

In the fifth chapter, we use the mod-p decomposition of lie groups for further calculation.Main introduction of this charpter is the results of mod-p decomposition of lie groups,and we also give a brief introduction of how to use the mod-p decomposition to do further calculate.