PL equivariant surgery and invariant decompositions of 3-manifolds

Abstract

Using normal surface theory [H,, J2], we introduce the notion of least weight normal surfaces. The weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface. If we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class, then we have least weight normal surfaces. It is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area (minimal) surfaces. They provide a piecewise linear (PL) environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces. In an impressive series of papers Meeks and Yau [M-Y i , M-Y z, M-Y 3, M-Y,, M-Y,], Meeks, Simon, and Yau [M-S-Y], Scott [S], Meeks and Scott [M-S], and Freedman, Hass, and Scott [F-H-S] introduced the analysis and geometry of least area surfaces into the study of topological questions about 3-manifolds. The consequences have led to the resolution of many outstanding problems in the topology of 3-manifolds. Since most of the problems being solved with these new techniques are topological in nature, it has been felt that there should be a more topological approach to the proofs of these results. Besides, the bulk of the work in the theory of