Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an elem...Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an element in H<sup>2</sup>（N,）, where is the local coefficient determined by w<sub>1</sub>（v<sub>f</sub>） =w<sub>1</sub>（N）. It is very important to determine e（v<sub>f</sub>）[N] for all embeddings. This problem is closely related to whether a two-dimensional homology class can be represented by a smooth embedded sphere. In this note, we determine all the possible normal Euler numbers of embedding real projective plane into indefinite 4-manifolds.展开更多
Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H&l...Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H<sub>2</sub>（M,（?））,where （?） denotes the twisted integer coefficients determined byw<sub>1</sub>（v）=w<sub>1</sub>（M）.We study the possible values of e（v）[M],and prove H<sub>1</sub>（N-M）=Z<sub>2</sub> or 0.Underthe condition of H<sub>1</sub>（N-M,Z）=Z<sub>2</sub>,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β<sub>2</sub>）,2σ（N）-2（n+β<sub>2</sub>-2）,2σ（N）-2（n+β<sub>2</sub>-4）,…,2σ（N）+2（n+β<sub>2</sub>）,where σ（N） is the usual index of N,n the nonorientable genus of M and β<sub>2</sub> the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S<sup>4</sup>.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.展开更多
A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In th...A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.展开更多
Let M be a closed oriented surface immersed in R4 . Associated it one has the generalized Gauss map from M into the Grassmann manifold G 4,2 . This note will be concerned with the geometry of the generalized Gauss map...Let M be a closed oriented surface immersed in R4 . Associated it one has the generalized Gauss map from M into the Grassmann manifold G 4,2 . This note will be concerned with the geometry of the generalized Gauss map by using the moving frame theory and the quaternion interpretation of Plcker coordinates. As one of consequences,we get the celebrated theorem of Chern and Spanier,Hoffman and Osserman,who proved it by quite different methods. At last,we give an explicit construction of a series of immersions of S2 in R4 with any given normal Euler number.展开更多
A foundamental problem in 4-dimensional differential topology iS to find a surface with minimal genus that can represent a given two-dimensional homology class. Hsiang, Rohlin have got some results about the homologic...A foundamental problem in 4-dimensional differential topology iS to find a surface with minimal genus that can represent a given two-dimensional homology class. Hsiang, Rohlin have got some results about the homological l-connected manifolds. This note will discuss this problem under the condition that H<sub>1</sub>（N<sup>4</sup>） is finite. Under this assumption, the possible normal Euler numbers of embedding nonorientable surfaces in 4-manifolds are also determined.展开更多
文摘Let N be a closed, nonorientable surface, M be a simply connected 4-manifold. f: N→M is an embedding with normal bundle v<sub>f</sub>. The normal Euler class e（v<sub>f</sub>） of f is an element in H<sup>2</sup>（N,）, where is the local coefficient determined by w<sub>1</sub>（v<sub>f</sub>） =w<sub>1</sub>（N）. It is very important to determine e（v<sub>f</sub>）[N] for all embeddings. This problem is closely related to whether a two-dimensional homology class can be represented by a smooth embedded sphere. In this note, we determine all the possible normal Euler numbers of embedding real projective plane into indefinite 4-manifolds.
文摘Let N be a closed,orientable 4-manifold satisfying H<sub>1</sub>（N,Z）=0,and M be a closed,connected,nonorientable surface embedded in N with normal bundle v.The Euler class e（v）ofv is an element of H<sub>2</sub>（M,（?））,where （?） denotes the twisted integer coefficients determined byw<sub>1</sub>（v）=w<sub>1</sub>（M）.We study the possible values of e（v）[M],and prove H<sub>1</sub>（N-M）=Z<sub>2</sub> or 0.Underthe condition of H<sub>1</sub>（N-M,Z）=Z<sub>2</sub>,we conclude that e（v）[M]can only take the followingvalues:2σ（N）-2（n+β<sub>2</sub>）,2σ（N）-2（n+β<sub>2</sub>-2）,2σ（N）-2（n+β<sub>2</sub>-4）,…,2σ（N）+2（n+β<sub>2</sub>）,where σ（N） is the usual index of N,n the nonorientable genus of M and β<sub>2</sub> the 2nd real Bettinumber.Finally,we show that these values can be actually attained by appropriate embeddingfor N=homological sphere.In the case of N=S<sup>4</sup>.this is just the well-known Whitney conjectureproved by W.S.Massey in 1969.
文摘A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.
基金supported by National Natural Science Foundation of China(Grant Nos. 10531090 and 10229101)the Chang Jiang Scholars Program
文摘Let M be a closed oriented surface immersed in R4 . Associated it one has the generalized Gauss map from M into the Grassmann manifold G 4,2 . This note will be concerned with the geometry of the generalized Gauss map by using the moving frame theory and the quaternion interpretation of Plcker coordinates. As one of consequences,we get the celebrated theorem of Chern and Spanier,Hoffman and Osserman,who proved it by quite different methods. At last,we give an explicit construction of a series of immersions of S2 in R4 with any given normal Euler number.
基金the National Natural Science Foundation of China.
文摘A foundamental problem in 4-dimensional differential topology iS to find a surface with minimal genus that can represent a given two-dimensional homology class. Hsiang, Rohlin have got some results about the homological l-connected manifolds. This note will discuss this problem under the condition that H<sub>1</sub>（N<sup>4</sup>） is finite. Under this assumption, the possible normal Euler numbers of embedding nonorientable surfaces in 4-manifolds are also determined.
