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.

References

[1]	J. Hempel, "3-manifolds" , Princeton Univ. Press (1976)
[2]	F. Waldhausen, "On irreducible 3-manifolds which are sufficiently large" Ann. of Math. , 87 (1968) pp. 56–88
[3]	W.H. Jaco, "Lectures on three-manifold topology" , Amer. Math. Soc. (1980)
[a1]	E.E. Moise, "Geometric topology in dimensions 2 and 3" , Springer (1977)