Mathematical Sciences: Studies in Geometric Topology

Quinn计划的最终目标是开发光滑紧致4流形的不变量，例如，可以检测光滑4维庞加莱猜想的反例。课程的第一部分（本项目）将主要致力于研究一个模型问题，关于2-络合物的Andrews-Curtis猜想。所考虑的不变量是Atiyah和Witten意义上的“拓扑量子场论”。庞加莱猜想断言，任何流形只要与拓扑球具有某些共同的简单拓扑性质，就一定是拓扑球。它最初是在世纪之交针对三维流形提出的，随后推广到所有更高的维度。奇怪的是，所有这些概括现在都已经解决了（肯定的），但三维球体的原始情况仍然抵制所有攻击。上述备注适用于拓扑流形。对于光滑流形，情况略有不同。这里庞加莱猜想的三维和四维版本都是开放的。Quinn希望通过量子场论的数学公式来解决四维空间的问题。

The eventual goal of Quinn's program is to develop invariants of smooth compact 4-manifolds which, for example, could detect counterexamples to the smooth 4-dimensional Poincare conjecture. The first part of the program (this project) will be devoted primarily to study of a model problem, the Andrews-Curtis conjecture on 2-complexes. The invariants under consideration are "topological quantum field theories" in the sense of Atiyah and Witten. The Poincare conjecture asserts that any manifold that has certain simple topological properties in common with a topological sphere must actually be a topological sphere. It was originally stated for three-dimensional manifolds around the turn of the century and subsequently generalized to all higher dimensions. Curiously, all these generalizations have now been settled (in the affirmative), but the original case for three-dimensional spheres still resists all assaults. The foregoing remarks apply to topological manifolds. The situation is slightly different for smooth manifolds. Here both the three- and the four-dimensional versions of the Poincare conjecture remain open. What Quinn is hoping to do is to settle the four-dimensional case, using methods inspired by a mathematical formulation of quantum field theory.