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We wish to study the geometry and topology of 3-manifolds. To this end we will need the central notion of spine of a 3-manifold. Indeed, we will be able to refine the notion of spine to get a class of spines that give us a natural presentation of 3-manifolds.

Denote by M the set of all compact 3-manifolds. We wish to study it systematically and comprehensively. The crucial question is the choice of filtration in M . It would be desirable to have a finite number of 3-manifolds in each term of the filtration, all of them being in some sense simpler than those in the subsequent terms. A useful tool here would be a measure of “complexity” of a 3-manifold. Given such a measure, we might hope to enumerate all “simple” manifolds before moving on to more complicated ones. There are several well-known candidates for such a complexity function. For example, take the Heegaard genus g ( M ), defined to be the minimal genus over all Heegaard decompositions of M . Other examples include the minimal number of simplices in a triangulation of M and the minimal crossing number in a surgery presentation for M .

Normal surfaces were introduced by Kneser in 1929 [66]. The theory of normal surfaces was further developed by W. Haken in the early 1960s [38]. Its fundamental importance to the algorithmic topology cannot be overestimated. Most of the work on 3-manifolds since then is based on or related to it.

The theory of normal surfaces is used extensively in algorithmic topology. Algorithms based on it most often follow the General Scheme described below. Suppose that we wish to solve a problem about a given 3-manifold M .

As we have mentioned earlier, recognizing irreducibility of 3-manifolds requires the existence of a recognition algorithm for the sphere S 3. Another motivation for constructing such an algorithm is the following. To the late 1970s topologists elaborated methods for proving an algorithmic classification theorem for Haken manifolds (though a complete proof appeared only in 1997, see Chap. 6). These methods play a crucial role in solving many other problems about Haken manifolds, but they do not work for manifolds which are not sufficiently large. What can one do with them?

In preceding chapters we described a number of important algorithms, which make heavy use of the Haken method of normal surfaces. As a rule, algorithms based on that method have exponential complexity and hence are impractical. In particular, although the recognition problem for Haken manifolds has an algorithmic solution, there is no chance of it being be realized by a computer program, at least in the foreseeable future. On the other hand, quite often experienced topologists recognize 3-manifolds rather quickly. They use other algorithms which, whether based on rigorous mathematics or intuition, are much more efficient. However, this gain does not come for free. The price is that one has to allow an algorithm to be only partial (i.e., not an algorithm at all in the formal meaning of this term). The problem of finding an efficient partial algorithm for answering a particular class of geometric questions is in itself a well-stated mathematical problem. Trying to solve it, we inevitably discover new structural properties of geometric objects.