Wen does Dehn surgery yield a Seifert fibered manifold?

W. Thurston的双曲手术定理表明，除了有限次的Dehn手术外，双曲结上的所有手术都产生双曲流形。因此，研究那些产生非双曲流形的特殊手术是很重要的。通过与Kimihiko Motegi（日本大学）的联合工作，作者研究了手术何时产生Seifer纤维流形或包含基本回路的流形，非双曲流形的典型例子，并获得了以下结果。已经知道，对于周期大于2的结，没有Dehn手术不会产生Seifert纤维歧管。对于周期为2的结，我们证明了如果这样的结产生一个塞弗特纤维流形，那么该结通过其周期映射的商是一个环面结。Dehn什么时候对双曲周期结进行手术得到包含基本环面的流形？我们证明了一个周期大于2的结当且仅当其格为1，周期为3，手术系数为0时允许进行这种手术；如果周期为2的结点允许这样的手术，则手术系数为整数。Dehn手术获得了哪些Seifert纤维流形？我们从Seifert纤维流形的异常纤维指标的角度研究了这个问题，并证明了：对于任意整数p， q， r具有对副素数，对双曲结进行手术得到一个具有3个异常纤维指标p， q， r的Seifert纤维流形。

W. Thurston's Hyperbolic Surgery Theorem shows that all but a finite number of Dehn surgeries on a hyperbolic knot yield hyperbolic manifolds. It is thus important to study those exceptional surgeries yielding non-hyperbolic manifolds. By the join work with Kimihiko Motegi (Nihon Univ.) the author studied when surgey yields a Seifer fibered manifold or a manifold containing essential tours, typical examples of non-hyperbolic manifolds, and obtained the following results.1. It had been already known that no Dehn surgery on a knot with period greater than 2 does not yield a Seifert fibered manifold. Regarding a knot with period 2, we showed that if such a knot yeilds a Seifert fibered manifold, then the quotient of the knot by its periodic map is a torus knot.2. When does Dehn surgery on a hyperbolic, periodic knot yield a manifold containing an essential torus? We showed that a knot with period greater than 2 admits such a surgery if and only if its genus is 1, its period is 3, and the surgery coefficient is 0; if a knot with period 2 admits such a surgery, then the surgery coefficient is integer.3. What Seifert fibered manifolds are obtined by Dehn surgery? We studied this problem in terms of indices of exceptinal fibers of Seifert fibered manifolds, and showed that : for arbitary integers p, q, r with some pair coprime, a surgery on some hyperbolic knot yields a Seifert fibered manifold with 3 exceptional fibers of indices p, q, r.