Studies on Combinatorial Structures of 3-Manifolds

(1)本文证明了表示同一闭3-流形的两个DS-图可以通过一系列“初等变形”相互转化。(2)关于Heegaard分裂本文发现了Heegaard分裂可约的一个新条件，这个条件用d-伪核的概念来描述。(3)本文证明了任何同伦3-球面都存在一个“非常特殊的框架链”，它具有很好的性质，并且与Heegaard分裂密切相关。(4)关于Poincare猜想利用上述两个结果（2）和（3），我们提出了一种新的方法来攻击著名的“Poincare猜想”。

(1) On Ds-diagramsIt was shown that two Ds-diagrams representing the same closed 3-manifold can be transformed to each other by a sequence of "elementary deformations".(2) On Heegaard splittingsA new condition for a Heegaard splitting to be reducible has been found. This condition is described by the notion of a "d-pseudo core".(3) On framed links in a homotopy 3-sphereIt was shown that any homotopy 3-sphere admits a "very special framed link" , which enjoys a good property and closely related to a Heegaard splitting.(4) On the Poincare conjectureUsing the above two results (2) and (3), we have proposed a new method for attacking the famous "Poincare conjecture".