Controlled Surgery

这一建议涉及两个主题的控制手术。第一个是同伦理论的方法来控制手术的庞加莱空间的4维及以上。这为同调流形的极限构造提供了输入。主要的新结论将是在4维中构造奇异的同调流形（Bryant-Ferry-Mio-Weinberger构造在6维及以上），并通过已知的分解定理证明了4维拓扑手术猜想。第二个主题是利用受控K理论的手术组的变化。这应该有更好的形式性质，例如应该更经常地满足Farrell-Jones同构猜想。庞加莱空间具有与流形相似的全局同调对偶。给定一个到度量空间的映射，我们可以测量对偶结构的“大小”：我们必须走多远才能找到对偶圈。流形具有尺寸为0的对偶性，因为对偶性来自局部结构。庞加莱空间一般没有约束，所以尺寸是控制空间的直径。该方案的关键部分是一个尺寸缩减过程：给定一个具有尺寸为λ的对偶的庞加莱空间，找到一个具有小得多的尺寸控制的等价庞加莱空间。所提议的办法应适用于4维及以上的空间。在这种情况下，它将为一个精心设计但已经建立的论点提供输入，以证明四维拓扑手术猜想。这个猜想是主要的未解决的问题，在拓扑4维。

This proposal concerns two topics in controlled surgery. The first is a homotoy-theoretic approach to controlled surgery on Poincare spaces of dimension 4 and greater. This provides input for a limit construction of homology manifolds. The principal new conclusion would be construction of exotic homology manifolds in dimension 4 (the Bryant-Ferry-Mio-Weinberger construction works in 6 and above) and, via a known resolution theorem, a proof of the 4-dimensional topological surgery conjecture. The second topic is a variation on surgery groups making use of controlled K-theory. This should have better formal properties, for instance should more often satisfy the Farrell-Jones isomorphism conjecture.This proposal concerns controlled surgery on Poincare spaces. Poincare spaces have global homological duality similar to that of manifolds. Given a map to a metric space one can measure the "size" of the duality structure: how far one must go to find dual cycles. Manifolds have duality with size 0 because duality follows from the local structure. Poincare spaces generally have no constraints, so the size is the diameter of the control space. The key part of the proposal is a size-reducing process: given a Poincare space with duality of size epsilon, find an equivalent Poincare space with much smaller size control. The approach proposed should work for spaces of dimension 4 and above. In this case it would provide input to an elaborate but already established argument to give a proof of the 4-dimensional topological surgery conjecture. This conjecture is the major unresolved problem in topology in dimension 4.