ON ELEMENTARY DEFORMATIONS OF MAPS OF SURFACES INTO 3-MANIFOLDS I
ON ELEMENTARY DEFORMATIONS OF MAPS OF SURFACES INTO 3-MANIFOLDS I
复制标题
关于三流形面图的基本变形 I
DOI：
--
复制
发表时间：
1985
期刊：
影响因子：
--
通讯作者：
T. Nagase
中科院分区：
文献类型：
--
作者：
Tatsuo Homma;T. Nagase
S. Smale [S] classified the set of immersions of $S^{2}$ into $E^{n}(2<n)$ by regular homotopy. M. Hirsch [Hr] generalized Smale’s result as follows: The regular homotopy classes of a $C^{\infty}$ manifold $M^{m}$ to a $C^{\infty}$ manifold $N^{n}(m>n)$ are in $one_{\wedge}to$ one correspondence with homotopy classes of bundle maps of tangent bundle $T(M^{m})$ to $T(N^{n})$ . This Smale-Hirsch theorem is a theorem about global moves of homotopy. To attack the Poincar\’e Conjecture, W. Haken [Hk] examined local moves between immersions of a 2-sphere into a homotopy 3-ball $N^{3}$ . What he found is that $\partial N^{S}$ can be deformed into a 3-ball in $N^{S}$ by four types of local deformations, elementary deformations, such that these four types of deformations take place in a special order. It is natural to consider the following question: how many kinds of elementary deformations are needed, for two given immersions which are regularly homotopic, to convert one immersion to the other? Or more generally, how many kinds of elementary deformations are needed to convert one nice map to another nice map? A nice map is a piecewise linear map of a surface into a 3-manifold whose singularities consist of a finite number of double curves, triple points and branch points. In this Paper we shall consider deformations of a homotopy between two nice maps of a surface in a 3-manifold into a finite sequence of elementary deformations. Elementary deformations are basic local moves of nice maps. This is the first of two Papers devoted to the study of deformations of a homotopy between two nice maps of a surface to a 3-manifold into a finite sequence of elementary deformations. In the second paper [HN], we shall prove the following:
S. Smale [S] classified the set of immersions of $S^{2}$ into $E^{n}(2<n)$ by regular homotopy. M. Hirsch [Hr] generalized Smale’s result as follows: The regular homotopy classes of a $C^{\infty}$ manifold $M^{m}$ to a $C^{\infty}$ manifold $N^{n}(m>n)$ are in $one_{\wedge}to$ one correspondence with homotopy classes of bundle maps of tangent bundle $T(M^{m})$ to $T(N^{n})$ . This Smale-Hirsch theorem is a theorem about global moves of homotopy. To attack the Poincar\’e Conjecture, W. Haken [Hk] examined local moves between immersions of a 2-sphere into a homotopy 3-ball $N^{3}$ . What he found is that $\partial N^{S}$ can be deformed into a 3-ball in $N^{S}$ by four types of local deformations, elementary deformations, such that these four types of deformations take place in a special order. It is natural to consider the following question: how many kinds of elementary deformations are needed, for two given immersions which are regularly homotopic, to convert one immersion to the other? Or more generally, how many kinds of elementary deformations are needed to convert one nice map to another nice map? A nice map is a piecewise linear map of a surface into a 3-manifold whose singularities consist of a finite number of double curves, triple points and branch points. In this Paper we shall consider deformations of a homotopy between two nice maps of a surface in a 3-manifold into a finite sequence of elementary deformations. Elementary deformations are basic local moves of nice maps. This is the first of two Papers devoted to the study of deformations of a homotopy between two nice maps of a surface to a 3-manifold into a finite sequence of elementary deformations. In the second paper [HN], we shall prove the following: