Three-dimensional manifold

A topological space each point of which has a neighbourhood homeomorphic to three-dimensional real space $ \mathbf R ^ {3} $ or to the closed half-space $ \mathbf R _ {+} ^ {3} $. This definition is usually supplemented by the requirement that a three-dimensional manifold as a topological space be Hausdorff and have a countable base. The boundary of a three-dimensional manifold, that is, its set of points that only have neighbourhoods of the second, rather than the first, of the above types, is a two-dimensional manifold without boundary. Methods of the topology of three-dimensional manifolds are very specific and therefore occupy a special place in the topology of manifolds.

Examples. Some properties of three-dimensional manifolds that, in general, do not hold in higher dimensions are: an orientable three-dimensional manifold is always parallelizable; a closed three-dimensional manifold bounds some four-dimensional manifold; one can always introduce into a three-dimensional manifold piecewise-linear and differentiable structures, and any homeomorphism between two three-dimensional manifolds can be approximated by a piecewise-linear homeomorphism as well as by a diffeomorphism.

One of the most widespread methods of describing a three-dimensional manifold is the use of Heegaard decompositions and the Heegaard diagrams closely related to them (cf. Heegaard decomposition; Heegaard diagram). The essence of this method is that any closed oriented three-dimensional manifold $ M $ can be decomposed into two submanifolds with a common boundary, each of which is homeomorphic to a standard complete pretzel (or handlebody, cf. Handle theory) $ V $ of some genus $ n $. In other words, a three-dimensional manifold $ M $ can be obtained by glueing two copies of a complete pretzel $ V $ along their boundaries by some homeomorphism. This fact enables one to reduce many problems in the topology of three-dimensional manifolds to those in the topology of surfaces. The smallest possible number $ n $ is called the genus of the three-dimensional manifold $ M $. Another useful method of describing a three-dimensional manifold is based on the existence of a close connection between three-dimensional manifolds and links in $ S ^ {3} $( cf. Knot theory): Any closed oriented three-dimensional manifold $ M $ can be represented in the form $ M = \partial W $, where the four-dimensional manifold $ W $ is obtained from the $ 4 $- ball $ B ^ {4} $ by attaching handles of index 2 along the components of some framed link $ L $ in $ S ^ {3} = \partial B ^ {4} $. Equivalently, a three-dimensional manifold $ M $ can be obtained from the sphere $ S ^ {3} $ by spherical surgery. It may be required in addition that all the components of the link $ L $ have even framings, and then the manifold $ W $ thus obtained is parallelizable. Often one uses the representation of a three-dimensional manifold as the space of a ramified covering of $ S ^ {3} $. If $ L $ is a link in $ S ^ {3} $, then any finitely-sheeted covering space of $ S _ {3} /L $ can be compactified by certain circles to give a closed three-dimensional manifold $ M $. The natural projection $ p: M \rightarrow S ^ {3} $, which is locally homeomorphic outside $ p ^ {-} 1 ( L) $, is called the ramified covering of $ S ^ {3} $ with ramification along $ L $. Any three-dimensional manifold of genus 2 is a double covering of the sphere with ramification along some link, while in the case of a three-dimensional manifold of arbitrary genus one can only guarantee the existence of a triple covering with ramification along some knot. This circumstance is the main cause why the three-dimensional Poincaré conjecture and the problem of the algorithmic recognition of a sphere have so far (1984) only been solved in the class of three-dimensional manifolds of genus 2.

The main problem in the topology of three-dimensional manifolds is that of their classification. A three-dimensional manifold $ M $ is said to be simple if $ M = M _ {1} \# M _ {2} $ implies that exactly one of the manifolds $ M _ {1} $, $ M _ {2} $ is a sphere. Every compact three-dimensional manifold decomposes into a connected sum of a finite number of simple three-dimensional manifolds. This decomposition is unique in the orientable case and is unique up to replacement of the direct product by $ S ^ {2} \widetilde \times S ^ {1} $ in the non-orientable case. Instead of the notion of a simple three-dimensional manifold, it is often more useful to use the notion of an irreducible three-dimensional manifold, that is, a manifold in which every $ 2 $- sphere bounds a ball. The class of irreducible three-dimensional manifolds differs from that of simple three-dimensional manifolds by just three manifolds: $ S ^ {3} $, $ S ^ {2} \times S ^ {1} $ and $ S ^ {2} \widetilde \times S ^ {1} $. Here the manifold $ S ^ {3} $ is irreducible, but is usually not considered to be simple, while the manifolds $ S ^ {2} \times S ^ {1} $ and $ S ^ {2} \widetilde \times S ^ {1} $ are simple but not irreducible. Irreducible three-dimensional manifolds with boundary have been fairly well studied. For example, any homotopy equivalence of pairs $ f: ( M, \partial M) \rightarrow ( M, \partial N) $, where $ M $, $ N $ are compact oriented irreducible three-dimensional manifolds with boundary, can be deformed into a homeomorphism. In the closed case it suffices for this that in addition the three-dimensional manifold $ M $ is sufficiently large, i.e. that it contains some two-sided incompressible surface. Here, a surface $ F \subset M $, $ F \neq S ^ {2} $, is said to be incompressible if the group homomorphism from $ \pi _ {1} ( F ) $ into $ \pi _ {1} ( M) $ induced by the imbedding is injective. If the first homology group of a compact irreducible three-dimensional manifold is infinite, then such a surface always exists. Any compact oriented irreducible sufficiently large three-dimensional manifold whose fundamental group contains an infinite cyclic normal subgroup is a Seifert manifold.

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