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
<正> This paper deals with problems of neat embeddings and immersions of compact n-manifolds with boundary into the disk Dn+k. In the metastable range, a sufficient condition of neat embeddings and immersions is...<正> This paper deals with problems of neat embeddings and immersions of compact n-manifolds with boundary into the disk Dn+k. In the metastable range, a sufficient condition of neat embeddings and immersions is given and representations of the set of neat isotopy classes of neat embeddings and the set of regular homotopy classes of immersions are obtained.展开更多
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
The purpose of this paper is to simplify the computations of the nor-mal bordism groups Ω<sub>i</sub>（W<sub>f</sub>,M×P<sup>∞</sup>;（?））and Ω<sub>i</sub>（C&...The purpose of this paper is to simplify the computations of the nor-mal bordism groups Ω<sub>i</sub>（W<sub>f</sub>,M×P<sup>∞</sup>;（?））and Ω<sub>i</sub>（C<sub>f</sub>,（?）w;θ<sub>f</sub>）which Salomonsenand Dax introduced respectively to study the existence and isotopy classificationof differential embeddings of manifolds in manifolds in the metastable range.Asimpler space pair（K<sub>f</sub>,M×P<sup>∞</sup>）is constructed to replace（W<sub>f</sub>,M×P<sup>∞</sup>）.It isshown that（K<sub>f</sub>,M×P<sup>∞</sup>）is homotopy equivalent to（W<sub>f</sub>,M×P<sup>∞</sup>）and homotopy（n-1）-equivalent to（C<sub>f</sub>,（?）W）.To demonstrate the efficacy of this simplification,the isotopy groups [M<sup>n</sup>（?）RP<sup>n+k</sup>],if n（?）2k-4 and M<sup>n</sup> is a closed（n-k+2）-connected manifold,and[M<sup>n</sup>（?）L（p;q<sub>1</sub>…,q<sub>m</sub>）],if 3n（?）4m-2,M<sup>n</sup> is a closed（2n-2m+1）-connected manifold and L is a （2m+1）-dimensional lens space,arespecifically computed.展开更多
In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R...In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R3, which have a formal embedding flows.展开更多
In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normaliza...In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems. Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity. Finally we pose some open problems on this subject.展开更多
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<正> This paper deals with problems of neat embeddings and immersions of compact n-manifolds with boundary into the disk Dn+k. In the metastable range, a sufficient condition of neat embeddings and immersions is given and representations of the set of neat isotopy classes of neat embeddings and the set of regular homotopy classes of immersions are obtained.
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The purpose of this paper is to simplify the computations of the nor-mal bordism groups Ω<sub>i</sub>（W<sub>f</sub>,M×P<sup>∞</sup>;（?））and Ω<sub>i</sub>（C<sub>f</sub>,（?）w;θ<sub>f</sub>）which Salomonsenand Dax introduced respectively to study the existence and isotopy classificationof differential embeddings of manifolds in manifolds in the metastable range.Asimpler space pair（K<sub>f</sub>,M×P<sup>∞</sup>）is constructed to replace（W<sub>f</sub>,M×P<sup>∞</sup>）.It isshown that（K<sub>f</sub>,M×P<sup>∞</sup>）is homotopy equivalent to（W<sub>f</sub>,M×P<sup>∞</sup>）and homotopy（n-1）-equivalent to（C<sub>f</sub>,（?）W）.To demonstrate the efficacy of this simplification,the isotopy groups [M<sup>n</sup>（?）RP<sup>n+k</sup>],if n（?）2k-4 and M<sup>n</sup> is a closed（n-k+2）-connected manifold,and[M<sup>n</sup>（?）L（p;q<sub>1</sub>…,q<sub>m</sub>）],if 3n（?）4m-2,M<sup>n</sup> is a closed（2n-2m+1）-connected manifold and L is a （2m+1）-dimensional lens space,arespecifically computed.
基金supported by NNSF of China grant 11271252by RFDP of Higher Education of China grant 20110073110054by FP7-PEOPLE-2012-IRSES-316338 of Europe
文摘In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R3, which have a formal embedding flows.
基金supported by the NNSF of China Grant 11271252the RFDP of Higher Education of China grant 20110073110054the FP7-PEOPLE-2012-IRSES-316338 of Europe
文摘In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems. Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity. Finally we pose some open problems on this subject.
