计算拓扑空间之间映射的同伦集是代数拓扑中的一个重要问题, 一般来说这是非常困难的. 设$X,Y$是两个拓扑空间, 奇异上同调函子$H^\ast$诱导出映射$H^\ast:\left[X,Y\right]\rightarrow Hom(H^*(Y),H^*(X))$.我们希望确定$Im\left(H^\ast\right)$，这是计算同伦集$\left[X,Y\right]$过程中的重要一步. 

对于紧单连通李群,王冉利用K-理论和陈特征给出了$Im(H^*)$的一个上界$H(Y,X)$. 对于紧单连通李群$G$, 记$DH(G)$为$H(G,G)$中所有对角同态的集合. 赵旭安给出了$DH(SU(n+1))$的一组基底. 本文采用类似的方法, 给出了$DH(Sp(n))$的一组基底. 利用这组基底, 本文证明了李群$Sp(2n)$上没有反定向自同胚. 本文的最后讨论了映射的实现问题.

Calculating the homotopy classes of maps between topological spaces is an important problem in algebraic topology, which is generally very difficult. We hope to get some constraints by studying homomorphisms between algebraic invariants. Let $X, Y$ be two topological spaces, the singular homology functor $H^\ast$ induces the function $H^\ast:\left[X,Y\right]\rightarrow Hom(H^*(Y),H^*(X))$. We want to determine $Im\left(H^\ast\right)$, which is an important step in calculating the homotopy set $\left[X,Y\right]$.

For a compact simply connected Lie group, using the knowledge of K-theory and Chern characteristic, Ran Wang has given an upper bound $H(Y,X)$ of $Im(H^*)$. For some common compact simply connected Lie groups $G$, we want to first compute the subset of  $H(G,G)$ which corresponding to diagonal maps, denoted as $DH(G)$, and try to find some rules. Xu-an Zhao has given a set of generators for $DH(SU(n+1))$. In this paper, I give a set of generators for $DH(Sp(n))$ using a similar method. As a corollary, I prove that there is no anti-oriented endomorphism on the Lie group $Sp(2n)$. At the end of this paper, I discussed the realization of the ring homomorphism in $DH(Sp(n))$.