Hakenness and b_1

Abstract

Three great theorems of Thurston read: Haken manifolds are hyperbolic; big ramified coverings are hyperbolic; big surgeries are hyperbolic. Recent developments indicate that the later two theorems are essentially a corollary of the first, that is there are much more Haken manifolds than expected by Thurston. In fact Freedman showed very recently that big ramified coverings are Haken. A version of his proof with various improvements was obtained by Cooper-Long and Cooper-Long-Reid. The two main contributions of the present paper are the following. I give an analytic proof of Freedman's result and the improved version of Cooper-Long-Reid. This proof is based on fundamentally different approach than Freedman's and is 10 times shorter, but uses the full forse of the hyperbolization theorem. Secondly, I prove that ANY ramified covering of a tight knot is Haken. Morover any knot becomes tight after a big surgery. So any ramified covering of a big surgery is Haken. Many other results of the paper are better seen from the introduction. The paper uses many different techniques and may be difficult to read for a beginner.