Haken manifold

sufficiently-large $3$-manifold, sufficiently-large three-dimensional manifold

A compact, $\mathbf P^2$-irreducible three-dimensional manifold which contains a properly embedded, incompressible, two-sided surface.

All objects and mappings are in the piecewise-linear category (cf. also Piecewise-linear topology). The surface $S^2$ denotes the two-dimensional sphere, while $\mathbf P^2$ denotes the projective plane. A surface $F$ properly embedded in a three-dimensional manifold $M$ is two-sided in $M$ if it separates its regular neighbourhood in $M$. A three-dimensional manifold $M$ is reducible (reducible with respect to connected sum decomposition) if it contains a properly embedded two-dimensional sphere that does not bound a three-dimensional cell in $M$. Otherwise, the three-dimensional manifold $M$ is irreducible. If the three-dimensional manifold $M$ is irreducible and does not contain an embedded, two-sided $\mathbf P^2$, it is said to be $\mathbf P^2$-irreducible. An orientable three-dimensional manifold is $\mathbf P^2$-irreducible if it is irreducible. A surface $F\neq S^2$ which is properly embedded in a three-dimensional manifold $M$ is compressible in $M$ if there is a disc $D$ embedded in $M$ such that $D\cap F=\partial D$ and the simple closed curve $\partial D$ does not bound a disc in $F$. Otherwise, such a surface $F$ is said to be incompressible in $M$. For two-sided surfaces it follows from the Dehn lemma that this geometric condition is equivalent to the inclusion mapping of fundamental groups, $\pi_1(F)\to\pi_1(M)$, being injective.

The three-dimensional cell is a Haken manifold, as is any compact, $\mathbf P^2$-irreducible three-dimensional manifold with non-empty boundary. In fact, a sufficient condition for a compact, $\mathbf P^2$-irreducible three-dimensional manifold $M$ to be a Haken manifold is that its first homology group with rational coefficients, $H_1(M,\mathbf Q)$, be non-zero. This condition is not necessary. Any closed three-dimensional manifold which is not a Haken manifold is said to be a non-Haken manifold. It is conjectured that every closed, $\mathbf P^2$-irreducible three-dimensional manifold with infinite fundamental group has a finite sheeted covering space (cf. also Covering surface) that is a Haken manifold.

An embedded, incompressible surface in a three-dimensional manifold is a very useful tool. However, in the case of three-dimensional manifolds with non-empty boundary, it is necessary to add an additional condition related to the boundary to achieve the same geometric character of the incompressible surface in a closed three-dimensional manifold. A properly embedded surface $F$ in a three-dimensional manifold $M$ with non-empty boundary is boundary compressible, written $\partial$-compressible, if there is a disc $D$ embedded in $M$ such that $\partial D$ is the union of two arcs $\alpha$ and $\beta$, $\alpha\cap\beta=\partial\alpha=\partial\beta$, $D\cap F=\alpha$, $D\cap\partial M=\beta$, and $\alpha$ does not cobound a disc in $F$ with an arc in $\partial F$. If a properly embedded surface $F$ in a three-dimensional manifold $M$ is not $\partial$-compressible, it is said to be boundary incompressible ($\partial$-incompressible). There is an algebraic interpretation of this geometric condition in terms of relative fundamental groups analogous to that given above in the case of an incompressible surface. Any compact, $\mathbf P^2$-irreducible three-dimensional manifold $M$ with non-empty boundary, other than the three-dimensional cell, contains a properly embedded, incompressible and $\partial$-incompressible surface that is not a disc parallel into $\partial M$.

Just as two-dimensional manifolds have families of embedded simple closed curves that split them into more simple pieces, the existence of incompressible and $\partial$-incompressible surfaces in Haken manifolds allow them to be split into more simple pieces. If $F$ is a properly embedded, two-sided surface in a three-dimensional manifold $M$ and $U(F)$ is the interior of some regular neighbourhood of $F$ in $M$, then $M'=M\setminus U(F)$ is the three-dimensional manifold obtained by splitting $M$ at $F$. A partial hierarchy for $M$ is a finite or infinite sequence of manifold pairs

$$(M_1,F_1),\ldots,(M_i,F_i),\ldots,(M_n,F_n),\ldots,$$

where $F_i$ is a properly embedded, two-sided, incompressible surface in $M_i$ which is not parallel into the boundary of $M_i$, and $M_{i+1}$ is obtained from $M_i$ by splitting $M_i$ at $F_i$. A partial hierarchy is said to be a hierarchy for $M$ if for some $n$, each component of $M_n$ is a a three-dimensional cell. Necessarily, a hierarchy for $M$ is a finite partial hierarchy, $(M_1,F_1),\ldots,(M_i,F_i),\ldots,(M_n,F_n)$, and $n$ is called the length of the hierarchy.

The fundamental theorem for Haken manifolds is that every Haken manifold has a hierarchy, $(M_1,F_1),\ldots,(M_i,F_i),\ldots,(M_n,F_n)$, where each $F_i$ is incompressible and $\partial$-incompressible in $M_i$. The existence of a hierarchy admits inductive methods of analysis and proof. This is exhibited in many of the major results regarding three-dimensional manifolds. For example, the problem of determining if two given three-dimensional manifolds are homeomorphic (the homeomorphism problem), the problem of determining if a given loop in a three-dimensional manifold is contractible (the word problem), and the problem of determining if a closed, orientable, irreducible, atoroidal (having no embedded, incompressible tori) three-dimensional manifold admits a metric with constant negative curvature, (uniformization), have all been solved for Haken manifolds but remain open (as of 2000) for closed, orientable, irreducible three-dimensional manifolds in general. The study of Haken manifolds is quite rich and includes a large part of the knowledge and understanding of three-dimensional manifolds.

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