In mathematics, a Haken manifold is a compact, P²-irreducible3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sidedincompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by Ian Agol.[1]
Haken manifolds were introduced by Wolfgang Haken (1961). Haken (1962) proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. William Jaco and Ulrich Oertel (1984) gave an algorithm to determine if a 3-manifold was Haken.
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
We will consider only the case of orientable Haken manifolds, as this simplifies the discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version of the surface, i.e. a trivial I-bundle. So the regular neighborhood is a 3-dimensional submanifold with boundary containing two copies of the surface.
Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M, resulting in M' . In effect, we've cut M along the surface S. (This is analogous, in one less dimension, to cutting a surface along a circle or arc.) It is a theorem that any orientable compact manifold with a boundary component that is not a sphere has an infinite first homology group, which implies that it has a properly embedded 2-sided non-separating incompressible surface, and so is again a Haken manifold. Thus, we can pick another incompressible surface in M' , and cut along that. If eventually this sequence of cutting results in a manifold whose pieces (or components) are just 3-balls, we call this sequence a hierarchy.
Applications
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, then it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e., incompressible. This makes proving the induction step feasible in many cases.
Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not. His outline was filled in by substantive efforts by Friedhelm Waldhausen, Klaus Johannson, Geoffrey Hemion, Sergeĭ Matveev, et al. Since there is an algorithm to check if a 3-manifold is Haken (cf. Jaco–Oertel), the basic problem of recognition of 3-manifolds can be considered to be solved for Haken manifolds.
Waldhausen (1968) proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed). So these three-manifolds are completely determined by their fundamental group. In addition, Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem; this is also true for virtually Haken manifolds.
The hierarchy played a crucial role in William Thurston's hyperbolization theorem for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.
^Agol, Ian (2013). "The virtual Haken conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning" (PDF). Documenta Mathematica. 18: 1045–1087. MR 3104553.
Haken, Wolfgang (1961). "Theorie der Normalflächen. Ein Isotopiekriterium für den Kreisknoten". Acta Mathematica. 105 (3–4): 245–375. doi:10.1007/BF02559591. ISSN 0001-5962. MR 0141106.
Haken, Wolfgang (1968). "Some results on surfaces in 3-manifolds". In Hilton, Peter J. (ed.). Studies in Modern Topology. Mathematical Association of America (distributed by Prentice-Hall, Englewood Cliffs, N.J.). pp. 39–98. ISBN978-0-88385-105-0. MR 0224071.
Jaco, William; Oertel, Ulrich (1984). "An algorithm to decide if a 3-manifold is a Haken manifold". Topology. 23 (2): 195–209. doi:10.1016/0040-9383(84)90039-9. ISSN 0040-9383. MR 0744850.
Johannson, Klaus (1979). "On the mapping class group of simple 3-manifolds". In Fenn, Roger A. (ed.). Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977). Lecture Notes in Mathematics. Vol. 722. Berlin, New York: Springer-Verlag. pp. 48–66. doi:10.1007/BFb0063189. ISBN978-3-540-09506-4. MR 0547454.
haken, manifold, mathematics, compact, irreducible, manifold, that, sufficiently, large, meaning, that, contains, properly, embedded, sided, incompressible, surface, sometimes, considers, only, orientable, which, case, compact, orientable, irreducible, manifol. In mathematics a Haken manifold is a compact P irreducible 3 manifold that is sufficiently large meaning that it contains a properly embedded two sided incompressible surface Sometimes one considers only orientable Haken manifolds in which case a Haken manifold is a compact orientable irreducible 3 manifold that contains an orientable incompressible surface A 3 manifold finitely covered by a Haken manifold is said to be virtually Haken The Virtually Haken conjecture asserts that every compact irreducible 3 manifold with infinite fundamental group is virtually Haken This conjecture was proven by Ian Agol 1 Haken manifolds were introduced by Wolfgang Haken 1961 Haken 1962 proved that Haken manifolds have a hierarchy where they can be split up into 3 balls along incompressible surfaces Haken also showed that there was a finite procedure to find an incompressible surface if the 3 manifold had one William Jaco and Ulrich Oertel 1984 gave an algorithm to determine if a 3 manifold was Haken Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms Contents 1 Haken hierarchy 2 Applications 3 Examples of Haken manifolds 4 See also 5 ReferencesHaken hierarchy EditWe will consider only the case of orientable Haken manifolds as this simplifies the discussion a regular neighborhood of an orientable surface in an orientable 3 manifold is just a thickened up version of the surface i e a trivial I bundle So the regular neighborhood is a 3 dimensional submanifold with boundary containing two copies of the surface Given an orientable Haken manifold M by definition it contains an orientable incompressible surface S Take the regular neighborhood of S and delete its interior from M resulting in M In effect we ve cut M along the surface S This is analogous in one less dimension to cutting a surface along a circle or arc It is a theorem that any orientable compact manifold with a boundary component that is not a sphere has an infinite first homology group which implies that it has a properly embedded 2 sided non separating incompressible surface and so is again a Haken manifold Thus we can pick another incompressible surface in M and cut along that If eventually this sequence of cutting results in a manifold whose pieces or components are just 3 balls we call this sequence a hierarchy Applications EditThe hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction One proves the theorem for 3 balls Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold then it is true for that Haken manifold The key here is that the cutting takes place along a surface that was very nice i e incompressible This makes proving the induction step feasible in many cases Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not His outline was filled in by substantive efforts by Friedhelm Waldhausen Klaus Johannson Geoffrey Hemion Sergeĭ Matveev et al Since there is an algorithm to check if a 3 manifold is Haken cf Jaco Oertel the basic problem of recognition of 3 manifolds can be considered to be solved for Haken manifolds Waldhausen 1968 proved that closed Haken manifolds are topologically rigid roughly any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism for the case of boundary a condition on peripheral structure is needed So these three manifolds are completely determined by their fundamental group In addition Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem this is also true for virtually Haken manifolds The hierarchy played a crucial role in William Thurston s hyperbolization theorem for Haken manifolds part of his revolutionary geometrization program for 3 manifolds Johannson 1979 proved that atoroidal anannular boundary irreducible Haken three manifolds have finite mapping class groups This result can be recovered from the combination of Mostow rigidity with Thurston s geometrization theorem Examples of Haken manifolds EditNote that some families of examples are contained in others Compact irreducible 3 manifolds with positive first Betti number Surface bundles over the circle this is a special case of the example above Link complements Most Seifert fiber spaces have many incompressible toriSee also EditManifold decomposition P2 irreducible manifoldReferences Edit Agol Ian 2013 The virtual Haken conjecture With an appendix by Agol Daniel Groves and Jason Manning PDF Documenta Mathematica 18 1045 1087 MR 3104553 Haken Wolfgang 1961 Theorie der Normalflachen Ein Isotopiekriterium fur den Kreisknoten Acta Mathematica 105 3 4 245 375 doi 10 1007 BF02559591 ISSN 0001 5962 MR 0141106 Haken Wolfgang 1968 Some results on surfaces in 3 manifolds In Hilton Peter J ed Studies in Modern Topology Mathematical Association of America distributed by Prentice Hall Englewood Cliffs N J pp 39 98 ISBN 978 0 88385 105 0 MR 0224071 Haken Wolfgang 1962 Uber das Homoomorphieproblem der 3 Mannigfaltigkeiten I Mathematische Zeitschrift 80 89 120 doi 10 1007 BF01162369 ISSN 0025 5874 MR 0160196 Hempel John 1976 3 manifolds Annals of Mathematics Studies Vol 86 Princeton University Press ISBN 978 0 8218 3695 8 MR 0415619 Jaco William Oertel Ulrich 1984 An algorithm to decide if a 3 manifold is a Haken manifold Topology 23 2 195 209 doi 10 1016 0040 9383 84 90039 9 ISSN 0040 9383 MR 0744850 Johannson Klaus 1979 On the mapping class group of simple 3 manifolds In Fenn Roger A ed Topology of low dimensional manifolds Proc Second Sussex Conf Chelwood Gate 1977 Lecture Notes in Mathematics Vol 722 Berlin New York Springer Verlag pp 48 66 doi 10 1007 BFb0063189 ISBN 978 3 540 09506 4 MR 0547454 Waldhausen Friedhelm 1968 On irreducible 3 manifolds which are sufficiently large Annals of Mathematics Second Series 87 1 56 88 doi 10 2307 1970594 ISSN 0003 486X JSTOR 1970594 MR 0224099 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