设 X 和 Y 是拓扑空间，对于上同调函子 H *，有 Boardman 映射 H *:[ X , Y ]→ Hom ( H *( Y ), H *( X ))。对于拓扑 K ﹣理论，考虑 Boardman 映射 K *:[ X , Y ]→ Hom Adams ( K *( Y ), K *( X ))，其中 HomAdams ( K *( Y ), K ( X )）是 K ( Y ）到 K *( X ）与 Adams 运算交换的同态的集合。
我们关注四元数 Grassmann 流形 Gk ( H n）的自映射的同伦集在映射 H *下的像。 Gk ( H n）的上同调环自同态 f 几乎都是 Adams 同态，即存在 I E Z ，对于任意 x E H4i( Gk ( H n))，都有 f ( x )= l i x 。记 deg ( f )= l 。通过陈特征 Ch : K *( X )→ H *( X ; Q ), Gk ( H n)的上同调环自同态 f 在 Hom Adams ( K *( Y ), K *( X )）中存在 fk = Ch -1fCh与其对应。若 f 可以实现为 G ( H n）的自映射，则必有 fk : K ( Gk ( H n))→ K ( Gk ( H n))是整系数环同态。这要求 deg ( f ）满足一些条件。
当 k =2, n =4,5,6,7时，我们计算了当 f 可实现时 deg ( f ）可能的取值，给出了 H *( G ( H n)）的自同态可实现的必要条件。这些计算有助于理解四元数 Grassmann 流形的自映射。

Let X and Y be topological spaces.There is a Boardman map H* : (XYHom(H*(Y), H*(X)) for the cohomology funtor H*. For K-theory, take Boardman map K*[X, Y] 一> Hom Adams(K*(Y), K*(X)) into account. Here Hom Adams(K*(Y), K*(X)) is the set of homomorphisms between K*(Y) and K*(X) which commute with the Adams operations.We concentrate on the image of the homotopy set of the self-maps of Grassmann manifoldsGk(Hn) under the map H*. Almost all endomorphisms of H*(Gk(Hn)) are Adams homomorphisms. That is, there exists l e Z, for all a e H4i(Gk (Hn)), f(a) = l. We denote deg( f) = lThere is a homomorphism fk = Ch-i fCh Ch Hom Adams(K*(Y), K*(X)) corresponding to f through Chern character Ch : K*(X) -> H*(X;Q). If f can be realized as a self-map ofGk(Hn), fk : K(Gk(Hn))一> K(G k(Hn)) must be a homomorphism with integral coefficients. For this, some conditions about deg( f) must be satisfied.
For k = 2, n = 4, 5, 6, 7, we calculate the possible values of deg( f) when f can be realized, which provides necessary conditions for endomorphisms of H*(Gk (Hn )) to be realized. Our work helps to understand the self-maps of quaternionic Grassmann manifolds.