m-维流形到(m+1)-维流形的零伦浸入的分类

NULL-HOMOTOPIC IMMERSIONS OF AN m-MANIFOLD IN AN (m+1)-MANIFOLD

这一节介绍一个引理和有关预备知识.对于拓扑空间 X、Y 而言,我们用[X,Y]表示 X 到 Y 的映射的同伦类组成的集合;[X,Y]'表示 X 到 Y 的保基点的映射的同伦类组成的集合.符号“(?)”表示(根据上下文)群同构或集合间的一一对应.

Let M and N be connected differentiable manifolds of dimension m and m+1respectively.Denote by [X,Y] the homotopy classes of maps from a topologicalspace X to another one Y,and [X,Y]′ the homotopy classe of pointed maps whenX,Y are pointed.If Y is a topological group,then [X,Y]=[X,Y]′ is understoodas a group.Denote by Imm[M,N]_c the regular homotopy classes of immersions ofM in N homotopic to a constant map c,and assame Imm[M,N]_c is nonenpty.Let T_(m+1)(N) be the total space of the O(m+1)-principal bundle on N associ-ated to the tangent bundle T(N),and SM the suspension of M.Then in the Puppeexact sequence[SM,N]′(?)[M,SO(m+1)](?)[M,T_(m+1)(N)]′→[M,N]′,(?) is a group homomorphism.Let (?)[SM,N] act on [M,SO(m+1)] as transla-tions.Identify Imm[M,R^(m+1)] with [M,SO(m+1)].Then the reflection of im-mersions induce an affine transformation (?) of the group [M,SO(m+1)].Denote byΓ the affine transformation group generated by (?)[SM,N]′ and (?).Theorem.If N is orientable,thenImm[M,N]_c(?)the quotient set [M,SO(m+1)]/(?)[SM,N]′;if N is non(?)orientable,thenImm [M,N]_c(?)the orbit set [M,SO(m+1)]/Γ of the action of Γ on [M,SO(m+1)].