Lusternik-Schnirelman 范畴是流形研究中的经典理论之一。虽然已经取得了一些研究成果，但是仍有大量重要问题没有得到解决。本项目主要研究三维流形的Lusternik-Schnirelman范畴及其各种推广，内容包括：三维流形的圆周范畴、环范畴、克莱因瓶范畴，以及与此密切相关的同调范畴和Singhof填充等。主要理论基础为一般化Seifert Fiber空间理论，即带反射曲线的Seifert Fiber空间理论。这是Seifert Fiber空间的自然推广，可以有效用于研究不可定向流形，使得问题的讨论更加统一完整。

Lusternik-Schnirelman Category is one of the classic theories in the study of manifold. Fruitful results have been obtained, but there are still a lot of open problems remained. This program focuses on Lusternik-Schnirelman Category and its generalization for 3-manifold, including S^1 Category、torus Category、Klein Bottle Category、Homological Category and the closely related Singhof fillings. The main theory to be used is generalized Seifert Fiber space theory, especially those with reflector curve. This is a natural generalization of Seifert Fiber space, and can be used to study non-orientable manifolds effectively and offers a more unifying discussion of problems.