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	Haken manifold -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		Haken manifolds are named after (additional info and facts about Wolfgang Haken) Wolfgang Haken, who pioneered the use of incompressible surfaces.
		We will consider only the case of (additional info and facts about orientable) 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.
		It is a theorem that cutting a Haken manifold along an incompressible surface results in a Haken manifold.
	www.absoluteastronomy.com /encyclopedia/H/Ha/Haken_manifold.htm (509 words)

	Haken manifold (Site not responding. Last check: 2007-10-22)
		In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided incompressible surface.
		Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces.
		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.
	www.worldhistory.com /wiki/H/Haken-manifold.htm (553 words)

	[No title]
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
		Orbifolds are manifolds with singularities such as reflection surfaces, where they resemble manifolds with boundary, and cone lines, where they are modelled (in the direction perpendicular to the cone line) by a cone with an angle of 360/n degrees for some n.
		PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	3-manifold - Wikipedia, the free encyclopedia
		In mathematics, a 3-manifold is a 3-dimensional manifold.
		The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds.
		One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case.
	en.wikipedia.org /wiki/3-manifold (346 words)

	Wolfgang Haken - TheBestLinks.com - Mathematician, Manifold, Topology, 1928, ... (Site not responding. Last check: 2007-10-22)
		Wolfgang Haken - TheBestLinks.com - Mathematician, Manifold, Topology, 1928,...
		Wolfgang Haken (born June 21, 1928) is a mathematician who specialized in topology, in particular 3-manifolds.
		He is best known as the co-solver of the four-color theorem, but has introduced several important ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces.
	www.thebestlinks.com /Wolfgang_Haken.html (186 words)

	Georgia Tech Geometry-Topology Seminar (Site not responding. Last check: 2007-10-22)
		Furthermore, we will show that these manifolds have very special properties --- that their Seiberg-Witten invariants are independent of the chamber structure, that they are not complex manifolds, and that they do not have metrics of positive scalar curvature.
		Haken manifolds are very well understood and most of the important questions in 3-manifold topology and geometry have been answered in this case.
		That such a manifold should always have a such a cover is the virtual Haken conjecture.
	www.math.gatech.edu /~stavros/gt.html (752 words)

	[No title]
		The resulting manifold still has a free fundamental group, so we continue inductively until we are left with a homotopy $3$-cell.
		Moreover, these manifolds are irreducible, since any PL $S^2$ embedded in a knot complement is embedded in $S^3$, and therefore bounds a $B^3$ on either side; consequently, any such $S^2$ bounds a $B^3$ on one side of the knot complement.
		Other examples are given by manifolds built from the same ideal tetrahedra, like the many manifolds (for example, the Whitehead link complement) obtained by identifying faces of the regular ideal octahedron.
	abel.math.harvard.edu /~tomc/qualfolder/probs.txt (3949 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		In the 1950's and 60's, when Papakyriakopoulos, Haken, Waldhausen, and others made advances in combinatorial 3-manifold topology, there was some faint hope that Heegaard splittings could lead to deep structure.
		I'm not sure, but I think that there is an example of a hyperbolic manifold for which the tree has only finitely many leaves (irreducible splittings).
		There is a theorem of Rubinstein and Scharlemann that for non-Haken manifolds, a linear number of stabilizations suffice.
	www.math.niu.edu /~rusin/known-math/99/3mflds (653 words)

	[No title]
		For example, Waldhausen proved in the 1960's that a Haken manifold M is determined up to homeomorphism by its fundamental group (`algebra determines topology').
		More recently Thurston proved the geometrization theorem, which states that every Haken manifold can be canonically split open along tori so that the pieces are geometric: each admits a metric modeled on one of 8 model geometries (`topology determines geometry').
		For example, since Gabai and Oertel proved that a laminar manifold has universal cover $\bf R^3$, Delman and Roberts' result immediately proves both the Property P Conjecture (non-trivial Dehn surgery never yields a simply-connected manifold) and the Cabling Conjecture (Dehn surgery on a non-cabled knot cannot yield a reducible manifold) for alternating knots.
	www.math.unl.edu /~mbrittenham2/personal/myresold.html (1253 words)

	iqexpand.com (Site not responding. Last check: 2007-10-22)
		In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups.
		The solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory.
		Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics.
	geometric_topology.iqexpand.com (539 words)

	4 Haken-like structures (Site not responding. Last check: 2007-10-22)
		In a sense, this was known before the work of Haken in the early 1960s, but it was only with Haken’s formulation of the incompressible surface that this principle was clearly realized.
		The basic content of the Waldhausen theorems is that the fundamental group of a Haken manifold determines its topological properties.
		Note that since a Haken manifold is irreducible, compact, with infinite fundamental group (which means its universal cover is non-compact), it is a K(pi,1), i.e.
	www.math.ucdavis.edu /~suh/3-manifolds/3-manifoldsse4.html (1724 words)

	Quasi-isometries preserve the geometric decomposition of Haken manifolds - Kapovich, Leeb (ResearchIndex)
		We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic.
		We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups.
		Kapovich and B. Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Vol.
	citeseer.ist.psu.edu /168446.html (629 words)

	Faculty in Mathematics & CS (Site not responding. Last check: 2007-10-22)
		These types of manifolds are known as Haken manifolds.
		These manifolds are well understood in terms of their geometry.
		Intuitively, and arithmetic manifolds has an associated group structure (the fundamental group) which inherits a lot of structure form a quaternion algebra.
	euler.slu.edu /Dept/Faculty/bart/research.html (556 words)

	5 “Virtual” Conjectures (Site not responding. Last check: 2007-10-22)
		If a manifold M is finitely covered by M’, with M’ having some property P, then we say M is virtually P. Perhaps surprisingly finite coverings of 3-manifolds are not well understood.
		It’s natural to ask how the topology of a 3-manifold is affected by various useful properties of a finite cover, and whether two given 3-manifolds have a common finite cover.
		Every closed, irreducible, orientable 3-manifold with infinite fundamental group is finitely covered by a Haken manifold.
	www.math.ucdavis.edu /~suh/3-manifolds/3-manifoldsse5.html (515 words)

	preprints
		We also show how our results imply the following: (1) a manifold that contains a non-separating surface contains an almost normal one, and (2) if a manifold contains a normal Heegaard surface then it contains two almost normal ones that are topologically parallel to it.
		We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions.
		Additionally, suppose $X \cup _F Y$ is a manifold obtained by gluing $X$ and $Y$, two connected small manifolds with incompressible boundary, along a closed surface $F$.
	pzacad.pitzer.edu /~dbachman/Papers/preprints.html (617 words)

	Citebase - Spaces of Incompressible Surfaces
		The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is contractible, except when the surface is the fiber of a surface bundle structure, in which case the space of all surfaces isotopic to the fiber has the homotopy type of a circle (the fibers).
		The main application from the 1976 paper is also rederived, the theorem (proved independently by Ivanov) that the diffeomorphism group of a Haken 3-manifold has contractible components, except in the case of certain Seifert manifolds when the components of the diffeomorphism group have the homotopy type of a circle or torus acting on the manifold.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9906074 (369 words)

	Cascade Topology Seminar Schedule (Site not responding. Last check: 2007-10-22)
		Suppose f is a map of a compact manifold to itself.
		However, there is ample evidence that they exist as holomorphic functions on the unit disk, that diverge everywhere on the unit circle but at roots of unity.
		When the manifold is a cylinder over a surface, this algebra quantizes the ring of SU(2) characters of the fundamental group of the surface.
	unr.edu /homepage/herald/schedule.html (790 words)

	Gromov, Lawson, Thurston: Hyperbolic 4-manifolds and conformally flat 3-manifolds
		GOLDMAN, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds, Trans.
		KAMISHIMA, Conformally flat manifolds whose developing maps are not surjective, I, Trans.
		GOLDMAN, Conformally flat manifolds whose developing maps are not surjective, II, to appear.
	www.numdam.org /numdam-bin/item?id=PMIHES_1988__68__27_0 (163 words)

	publications GDR tresses
		The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken.
		First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds.
		In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken.
	www.math.unicaen.fr /~picantin/GDRpub/GDRpub2002.html (1281 words)

	AGT 3 (2003) Paper 12 (Abstract) (Site not responding. Last check: 2007-10-22)
		A criterion for homeomorphism between closed Haken manifolds
		In this paper we consider two connected closed Haken manifolds denoted by M^3 and N^3, with the same Gromov simplicial volume.
		We give a simple homological criterion to decide when a given map f: M^3-->N^3 between M^3 and N^3 can be changed by a homotopy to a homeomorphism.
	www.univie.ac.at /EMIS/journals/AGT/AGTVol3/agt-3-12.abs.html (120 words)

	[No title]
		For an orientable Haken $3$-manifold $M$, Jaco and Shalen \cite{JS} and Johannson \cite{JO} proved that there is a family $\mathcal{T}$ of disjoint essential annuli and tori embedded in $M$, unique up to isotopy, and with the following properties.
		The fundamental group $G$ of $M$ is the fundamental group of a graph $\Gamma$ of groups, whose underlying graph is dual to the frontier of $V(M)$.
		When $G$ is the fundamental group of a Haken manifold $M$, the $V_{0}% $-vertices of $\Gamma_{1}$ essentially correspond to the peripheral components of the characteristic submanifold $V(M)$ of $M$.
	www.ii.uj.edu.pl /EMIS/journals/ERA-AMS/2002-01-003/2002-01-003.tex.html (3336 words)

	Citebase - The Farrell-Jones isomorphism conjecture for 3-manifold groups
		Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups.
		By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus \geq 2.
		In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0405211 (1521 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Title : 3-Manifolds and Geometric Group Theory Abstract : Abstract: The proposer plans to investigate purely algebraic analogs of the theory of the characteristic submanifold of a Haken 3-manifold.
		Some results in this direction have already been obtained by several authors but their uniqueness results are much weaker than that in the 3-manifold context.
		In this proposal, it is planned to investigate further the properties of this decomposition into geometric pieces.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9971555.txt (160 words)

	Atlas: The characteristic submanifold of a Haken 3-manifold by Peter Scott (Site not responding. Last check: 2007-10-22)
		Atlas: The characteristic submanifold of a Haken 3-manifold by Peter Scott
		In the 1970's, Jaco and Shalen, and Johannson, showed that any Haken 3-manifold with incompressible boundary has a canonical submanifold, called the characteristic submanifold.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cady-21.
	atlas-conferences.com /c/a/d/y/21.htm (91 words)

	Atlas: Canonical splittings of groups and 3-manifolds by Peter Scott (Site not responding. Last check: 2007-10-22)
		In the mid 1970's, Jaco and Shalen, and independently Johannson, proved that Haken 3-manifolds have a canonical submanifold (possibly empty) called the characteristic submanifold, and they showed that this submanifold has some important properties.
		We obtain new algebraic results which yield the topological result when applied to the fundamental group of a Haken 3-manifold.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagh-36.
	atlas-conferences.com /c/a/g/h/36.htm (170 words)

	published
		In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex.
		Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface.
		Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface.
	pzacad.pitzer.edu /~dbachman/Papers/published.html (687 words)

	AMCA: Smallish knots and triangulations by Marc Culler (Site not responding. Last check: 2007-10-22)
		It would follow from a difficult conjecture due to Lopez that non-Haken manifolds contain smallish knots.
		We will discuss a result which shows that certain simplification operations are possible when no edge of a 1-vertex triangulation of a non-Haken manifold is a smallish knot.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/g/h/75.htm (144 words)

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