一致嵌入的局部和末端变形定理
Local and end deformation theorems for uniform embeddings
摘要：
给出了度量流形（LD）中一致嵌入的一个局部形变性质，并从形式上研究了它的性质。证明了任何具有几何群作用的度量流形、典型度量空间（欧几里得空间、双曲空间和圆柱空间）以及对于κ \leq 0，κ-锥在任何紧致Lipschitz度量流形上的终结都具有（LD）性质.我们还制定了一个概念的终端变形性质的均匀嵌入在适当的产品结束（ED）。例如，紧致度量流形上的0-锥端具有性质（ED），如果它具有性质（LD）。证明了若度量流形M有2/3个具有性质（ED）的真积端，则M的有界一致同胚群具有一致拓扑，且在这些端上有单位的有界一致同胚子群上存在强形变收缩.我们还研究了一致同胚的变形中一致合痕的作用，并证明了κ-锥中的Alexander合痕诱导有界一致同胚群的某些子群的压缩.
英文摘要：
A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over any compact Lipschitz metric manifolds, all of them have the property (LD). We also formulate a notion of end deformation property for uniform embeddings over proper product ends (ED). For example, the 0-cone end over a compact metric manifold has the property (ED) if it has the property (LD). It is shown that if a metric manifold M has finitely many proper product ends with the property (ED), then the group of bounded uniform homeomorphisms of M endowed with the uniform topology admits a strong deformation retraction onto the subgroup of bounded uniform homeomorphisms which are identity over those ends. We also study a role of uniform isotopies in deformation of uniform homeomorphisms and show that Alexander isotopies in κ-cones induce contractions of some subgroups of groups of bounded uniform homeomorphisms.
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