Unrecognizability of Manifolds

Abstract

Abstract: One of the fundamental problems of topology is to decide, given two topological spaces, whether or not they are homeomorphic. This problem is known as the Homeomorphism Problem. To effectively answer this question one must first specify how a manifold is described and be sure that such a description is suitable for input into a computing device. The next step will be to come up with a general effective procedure to answer this question when applied to a sufficiently general class of spaces, of a specific dimension n 3 (PL manifolds, smooth manifolds, etc.) In this generality, it turns out that for compact PL manifolds, the homeomorphism problem is undecidable for spaces of dimension n 4, as was proved by A.A Markov (Mar58). Furthermore, a dramatic improvement of the previous result was discovered by S. P. Novikov (VKF74, pg.169) in the sense that for n 5, it is impossible to recognize the n-sphere, and in fact the same holds for any compact n-dimensional smooth manifold. The main purpose of this monograph is to present, for those readers with a basic background in algebra and topology, a detailed and accessible proof of S.P. Novikov's result. We follow the exposition that appears in the appendix of (Nab95). As a guide to the reader we offer an outline of the main points developed in our treatment. First, we prove the algorithmic unrecognizability of the n-sphere for n 5, according to the following steps: 1. We start from a finite presentation of a group G with unsolvable word problem. 2. Using the presentation for G we build a sequence of finitely presented groups {Gi} such that {Gi} is an Adian-Rabin sequence. 3. Following Novikov, we modify the sequence {Gi} and obtain a new sequence of finitely presented groups {G’i} which have trivial first and second homology, such that {G’i} is an Adian-Rabin sequence, i.e., we obtain a Novikov sequence. 4. Next, we construct a sequence of compact non-singular algebraic hypersurfaces Si C Rn+1, so that Si is a homology sphere and (pi)1(Si) = G’i. Moreover this is done in such a way that Si is diffeomorphic to Sn if and only if G’i is trivial. (From The Generalized Poincaré Conjecture and The Characterization of the smooth n-disc Dn, n6.) 5. Finally, arguing by contradiction, we assume that the n-sphere is algorithmically recognizable. Thus, if we apply this presumed algorithm to the elements of the sequence {Si} we could determine which of them are diffeomorphic to the n-sphere. This in turn would allow us to single out the trivial elements of the given Novikov sequence, but this is clearly impossible. As a final step, we apply the previous result to stablish the unrecognizability of the compact smooth n-manifolds, n 5, according to the following steps: 1. Assume for simplicity that M0 is a connected n-dimensional manifold that can be effectively recognized among the class of all compact n-dimensional manifolds. 2. Fix a compact n-dimensional manifoldM effectively generated from a Novikov sequence of groups and define M1 = M0#M. 3. Apply to M1 the procedure to recognize M0. 4. If the answer is No, then M is not a sphere. 5. If the answer is Yes, note that (pi)1(M) = 1 and then conclude that M is the sphere, since the only simply connected n-manifold generated from a Novikov sequence is the n-sphere. 6. From 4 and 5, an effective procedure to recognize M0 will allow us to recognize the n-sphere, which is a contradiction.