对于拓扑空间 和 , 确定映射同伦集 [, ] 是代数拓扑领域的重 要问题. 对于奇异上同调和拓扑 - 理论, 我们有 映射 ∗ ∶ [, ] → (∗ ( ), ∗ ()) 和 ∗ ∶ [, ] → (∗ ( ), ∗ ()), 其 中 (∗ ( ), ∗ ()) 表示 ∗ ( ) 到 ∗ () 与 Adams 运算交换的同态集 合。

给定 Stiefel 流形 (ℍ ) 的一个连续自映射 ，它诱导了奇异上同调环的自 同态 ，这个同态在有理陈特征下 ℎ ⊗ ℚ ∶ ∗ ( (ℍ )) ⊗ ℚ → ∗ ( (ℍ ); ℚ) 对应了 (∗ ( (ℍ )), ∗ ( (ℍ )) 中的 = ℎ−1ℎ. 利用 是一 个整系数同态这一事实，我们可以得到 Stiefel 流形的整系数上同调自同态可以 被连续映射实现的条件.

本文首先计算了 (ℍ ) 的 - 理论，然后以此为基础计算了它的陈特征，并 给出了可以被连续自映射实现的 的线性部分需要满足的条件. 我们计算了一 些具体的例子，并利用 Smith 分解简化了计算结果.

For topological spaces and ，to determine the set of homomorphisms [, ] is a big question in the field of algebraic topology.For the singular cohomology and -theory，we have Boardman maps ∗ ∶ [, ] → (∗ ( ), ∗ ()) and ∗ ∶ [, ] → (∗ ( ), ∗ ()) . Here (∗ ( ), ∗ ()) is the set of homomorphisms between ∗ ( ) and ∗ () which commute with the Adams operations.

Giving a continuous self-map of quaternionic Stiefel manifolds (ℍ ) ，it induces a self-homomorphism of ∗ ( (ℍ )).We can find a = ℎ−1ℎ in (∗ ( (ℍ )), ∗ ( (ℍ )) because of the Chern character ℎ ⊗ ℚ ∶ ∗ ( (ℍ )) ⊗ ℚ → ∗ ( (ℍ ); ℚ).Using the fact that is a homomorphism with integral coefficients，we can get some conditions which are needed to make the homomorphism could be realized by a continuous self-map .

In this paper ，we calculate the -theory of (ℍ ) first，then calculate the Chern character of it，and give the conditions that the linear part of which can be realized by continuous self-map needs to satisfy .We give some examples ，and use Smithdecomposition to simplify the results.