Mathematical Sciences: Topological Embeddings in Piecewise Linear Manifolds

Venema将研究流形的性质，包括拓扑流形和更受限制的分段线性(PL)流形，特别强调拓扑嵌入到分段线性流形中。(1)设X是PL_4-流形M的紧子集，在什么条件下X有可折叠为一维脊的任意闭邻域？(2)设F是闭曲面，W具有F的同伦型，是否存在F的拓扑嵌入W是同伦等价的？(3)在什么自然附加假设下，可以证明嵌入流形中的两个紧集有同胚补当且仅当它们具有相同的形状？(4)逼近定理对k-胞元以外的任何流形的余维二嵌入都有效吗？这些问题和其他问题指出了我们在理解非常自然的几何对象方面的基本缺陷，即拓扑和PL-流形，即使在低维的情况下也是如此。这样的流形出现在物理学中，就像我们生活的4维时空中的常微分方程组的解一样。

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.