对合的分解

Using the notion of biconnected sum we define the biconnected sum (T₁, M₁)§(T₂,M₂) of two involutions (T1M1) and (T2,M2) which is an involution on the biconnected sum M₁,§M₂. A connected involution is said to be reducible if it can be expressed as a biconnected sum of two connected involutions.Theorem Each connected involution (T, M) can be decomposed into a bi-connected sum of connected irreducible involutions (T, M)=(T₁, M₁)§…§(T_q,M_q),and (?) where the coefficients of H_{n_1}(M) are in Z/2 Z if M is unoriented, in Z if is oriented .     

EMBEDDED 2-SPHERES IN INDEFINITE 4-MANIFOLDS

51.IntroductionOne0ftheinterestingproblemsneartheheartof4dimensionaltopologyistodecidewhich2-dimensionalhomology/homotoPyclasscanberepresentedbyasmo0thlyembedded2-sphereinagivensmooth4-dimensionalmanifold.ItwasRohlin[19]wh0pointed0utintheearly195Ostousthatnotevery2-dimensionalhthmology/hom0topyclasscanbesorepresented.TheimportanceoftheproblemwasexplainedbyKervaireandMilnorl11].Onedecadelater,theystartedthestudyin1961,followedbyBoardm..I1]?Wall[25]5Tri.t..m[23]tHsiangandSzczarba[1o],andRoh…     

On the notion of cobordism for bounded manifolds

The usual concept of cobordism is concerned with closed manifolds. In this paper, we generalise the concept of cobordism to the extent of bounded manifolds and obtain a homotopy theorem similar to the theorem of Thom. Supported by the National Natural Science Foundation of China and partially supported by a grant of Zhejiang Natural Science Foundation.     

流形的双连通和分解

所论流形均假设为无边微分或PL紧致流形。设M_1和M_2为两个n维流形,从M_1,M_2各挖去两个不相交的n维闭胞腔之内部,而得M_1,M_2。然后将M_1的边缘与M_2的边缘用一个同胚来粘合,而得一新的n维流形,称为M_1和M_2的双连通和,记作M_1§M_2。一个流形若可表示为两个连通流形的双连通和,则称为可约化的;否则称为不可约化的。     

对合的分解

利用流形的双连通和概念可以定义两个n维流形上的对合(T_1,M_1)和(T_2,M_2)的双连通和,记作(T_1,M_1)§(T_2,M_2),它是定义在M_1与M_2的双连通和M_1§M_2上的一个对合T_1§T_2。一个对合称为可约化的,若它可以表为两个连通流形上的对合之双连通和。     

ON A PROBLEM OF WHITNEY

In 1944 H.Whitney raised a problem:Let M be an open smooth n-manifold.Doesthere exist an imbedding of M into R~(2n) with no limit point set?Introducing a sort of Morsenumber for open manifolds and using Whitney trick,the author gives a direct proof ofthe affirmative answer to it.     

GENERALIZATION OF THE NOTION OF FUNDAMENTAL GROUP

The classical definition of fundamental group for a topological space is based on the pathwise connectedness. A space with less path will not be able to be described effectively by its fundamental group. The author introduces a definition of generalized fundamental group for a given topological space by means of its own connectedness. For a well-behaved space, e.g., a locally pathwise and semilocally simply connected compact metric space, the generalized fundamental group coincides with the classical one.