Mathematical Sciences: Laminations, Representations and the Topology of 3-Manifolds

这个项目致力于流形和细胞复形，目的是探索3-流形的拓扑和3-流形所具有的层和表示的类型之间的相互关系。本文主要研究的问题有：(1)给定一个紧致定向叠层三维流形N和一个与N等价的不可约三维流形M同伦，M是否同胚于N？如果一个从M到它自己的同胚与这个恒等式同伦，它是否与这个恒等式同伦？(2)什么3-流形是层叠的？(3)计算3-球面中纽结的退化不变量。给出3-球体中的一个定向连杆，找到一种有效的方法来确定它是否是纤维的，如果是，则完成它的单轴运动。能否通过对纽结的运算得到Poincare猜想的反例？(4)计算Brieskorn同调球面表示空间的拓扑型。(5)定义并计算等变瞬子同调。(6)计算了同调球面上两个平面连接的相对Floer指数。(7)证明了4-球面上两个分支的环是零同伦的。

The goal of this project, devoted to manifolds and cell complexes, is to explore the interrelationships between the topology of 3-manifolds and the types of laminations and representations that 3-manifolds possess. The main questions to be investigated are the following: (1) Given a compact oriented laminated 3-manifold N and an irreducible 3-manifold M homotopy equivalent to N, is M homeomorphic to N? If a homeomorphism from M to itself is homotopic to the identity, is it isotopic to the identity? (2) What 3-manifolds are laminated? (3) Calculate the degenerate invariant of a knot in the 3-sphere. Given an oriented link in the 3-sphere, find an effective way to decide if it is fibred and if so complete its monodromy. Can one obtain a counterexample to the Poincare conjecture by surgery on a knot? (4) Calculate the topological type of the representation spaces of the Brieskorn homology spheres. (5) Define and calculate equivariant instanton homology. (6) Calculate the relative Floer index of two flat connections on homology spheres obtained by surgery on knots. (7) Show that a link of two components in the 4-sphere is nullhomotopic.