View source for Haken manifold

''sufficiently-large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300102.png" />-manifold, sufficiently-large three-dimensional manifold''

A compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300103.png" />-irreducible [[Three-dimensional manifold|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|Piecewise-linear topology]]). The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300104.png" /> denotes the two-dimensional sphere, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300105.png" /> denotes the [[Projective plane|projective plane]]. A surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300106.png" /> properly embedded in a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300107.png" /> is two-sided in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300108.png" /> if it separates its regular neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h1300109.png" />. A three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001010.png" /> is reducible (reducible with respect to [[Connected sum|connected sum]] decomposition) if it contains a properly embedded two-dimensional sphere that does not bound a three-dimensional cell in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001011.png" />. Otherwise, the three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001012.png" /> is irreducible. If the three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001013.png" /> is irreducible and does not contain an embedded, two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001014.png" />, it is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001016.png" />-irreducible. An orientable three-dimensional manifold is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001017.png" />-irreducible if it is irreducible. A surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001018.png" /> which is properly embedded in a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001019.png" /> is compressible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001020.png" /> if there is a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001021.png" /> embedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001023.png" /> and the simple closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001024.png" /> does not bound a disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001025.png" />. Otherwise, such a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001026.png" /> is said to be incompressible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001027.png" />. For two-sided surfaces it follows from the [[Dehn lemma|Dehn lemma]] that this geometric condition is equivalent to the inclusion mapping of fundamental groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001028.png" />, being injective.

The three-dimensional cell is a Haken manifold, as is any compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001029.png" />-irreducible three-dimensional manifold with non-empty boundary. In fact, a sufficient condition for a compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001030.png" />-irreducible three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001031.png" /> to be a Haken manifold is that its first [[Homology group|homology group]] with rational coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001032.png" />, 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001033.png" />-irreducible three-dimensional manifold with infinite [[Fundamental group|fundamental group]] has a finite sheeted covering space (cf. also [[Covering surface|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001034.png" /> in a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001035.png" /> with non-empty boundary is boundary compressible, written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001037.png" />-compressible, if there is a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001038.png" /> embedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001040.png" /> is the union of two arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001046.png" /> does not cobound a disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001047.png" /> with an arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001048.png" />. If a properly embedded surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001049.png" /> in a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001050.png" /> is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001051.png" />-compressible, it is said to be boundary incompressible (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001053.png" />-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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001054.png" />-irreducible three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001055.png" /> with non-empty boundary, other than the three-dimensional cell, contains a properly embedded, incompressible and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001056.png" />-incompressible surface that is not a disc parallel into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001057.png" />.

Just as two-dimensional manifolds have families of embedded simple closed curves that split them into more simple pieces, the existence of incompressible and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001058.png" />-incompressible surfaces in Haken manifolds allow them to be split into more simple pieces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001059.png" /> is a properly embedded, two-sided surface in a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001061.png" /> is the interior of some regular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001063.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001064.png" /> is the three-dimensional manifold obtained by splitting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001065.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001066.png" />. A partial hierarchy for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001067.png" /> is a finite or infinite sequence of manifold pairs

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001068.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001069.png" /> is a properly embedded, two-sided, incompressible surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001070.png" /> which is not parallel into the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001071.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001072.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001073.png" /> by splitting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001074.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001075.png" />. A partial hierarchy is said to be a hierarchy for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001076.png" /> if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001077.png" />, each component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001078.png" /> is a a three-dimensional cell. Necessarily, a hierarchy for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001079.png" /> is a finite partial hierarchy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001081.png" /> is called the length of the hierarchy.

The fundamental theorem for Haken manifolds is that every Haken manifold has a hierarchy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001082.png" />, where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001083.png" /> is incompressible and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001084.png" />-incompressible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001085.png" />. 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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Haken,   "Theorie der Normal Flächen I"  ''Acta Math.'' , '''105'''  (1961)  pp. 245–375</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Waldhausen,   "On irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130010/h13001086.png" />-manifolds which are sufficiently large"  ''Ann. of Math.'' , '''87'''  (1968)  pp. 56–88</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Thurston,   "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry"  ''Bull. Amer. Math. Soc. (N.S.)'' , '''6'''  (1982)  pp. 357–381</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Waldhausen,   "The word problem in fundamental groups of sufficiently large irreducible 3-manifolds"  ''Ann. of Math.'' , '''88'''  (1968)  pp. 272–280</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.T. Fomenko,   S.V. Matveev,   "Algorithmic and computer methods for three-manifolds" , Kluwer Acad. Publ.  (1997)</TD></TR></table>