Mathematical Sciences: Topological Embeddings in Piecewise Linear Manifolds

数学科学：分段线性流形中的拓扑嵌入

• 批准号: 8900822
• 负责人: Gerard Venema
• 金额: $ 3.3万
• 依托单位: Calvin University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8900822&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Embeddings; Piecewise

Venema will investigate properties of manifolds, both topological manifolds and the more restricted class of piecewise linear (PL) manifolds, with special emphasis on topological imbeddings into PL 4-manifolds. (1) Suppose X is a compact subset of the PL 4-manifold M. Under what conditions does X have arbitrarily close neighborhoods which collapse to 1-dimensional spines? (2) Suppose F is a closed surface and W has the homotopy type of F. Does there exist a topological imbedding of F into W which is a homotopy equivalence? (3) Under what natural additional hypotheses can one prove that two compacta imbedded in a manifold have homeomorphic complements if and only if they have the same shape? (4) Is the approximation theorem valid for codimension two imbeddings of any manifolds other than k-cells? These questions and others point to basic deficiencies in our understanding of very natural geometric objects, namely topological and PL-manifolds, even in low dimensions. Such manifolds arise in physics as the solutions of ordinary and differential equations set in the 4-dimensional space-time in which we live.

维内马将调查性质的流形， 拓扑流形和更严格的 分段线性（PL）流形，特别强调 拓扑嵌入到PL 4-流形中 (1)设X是 PL 4-流形M的紧子集。 在什么条件下 X是否有任意紧密的邻域， 1-次元脊椎 (2)设F是闭曲面，W 具有F的同伦类型。 是否存在一个拓扑 F嵌入W是同伦等价吗？ （三） 在什么样的自然附加假设下，人们可以证明两个 嵌入流形中的有同胚补的流形 当且仅当它们的形状相同 (4)是 余维二嵌入的逼近定理 除了k细胞还有其他的流形吗 这些问题和其他问题指出了 我们对非常自然的几何物体的理解， 拓扑和PL流形，即使在低维。 等 流形在物理学中是作为普通和 四维时空中的微分方程组， 我们生活的地方。