Hook manifold

In mathematics , hook manifolds are 3-dimensional manifolds that can be cut into simple pieces along incompressible surfaces and are therefore accessible to algorithmic treatment. They are named after Wolfgang Haken .

definition

A hook-manifold is a compact 3-manifold that - is irreducible and contains an ( actually embedded and bilateral ) incompressible surface . ${\displaystyle P^{2}}$

Explanations:

• A 3-manifold is irreducible if every embedded 2-sphere borders an embedded 3-sphere. It is irreducible if it is irreducible and does not contain a projective plane embedded on both sides. If it is orientable , then -irreducibility already follows from irreducibility. ${\displaystyle P^{2}}$ ${\displaystyle M}$ ${\displaystyle P^{2}}$
• If it has a non-empty edge, the incompressible surface should also be edge-incompressible . ${\displaystyle M}$

Examples

• The 3-dimensional full sphere is a hook manifold. ${\displaystyle B^{3}}$
• Every irreducible 3-manifold with a positive 1st Betti number ${\displaystyle M}$

${\displaystyle b_{1}(M)>0}$

is a hook manifold: because of Poincaré duality follows and one can show that a nontrivial homology class can be represented by an incompressible surface. In particular, every irreducible 3- manifold with a boundary is a hook-type manifold, e.g. every nodal complement . ${\displaystyle H_{2}(M,\mathbb {Z} )\not =0}$

• Almost all stretch surgeries on the figure eight knot result in manifolds that are not hooks. On the other hand, with this construction one gets some examples of hook manifolds with . ${\displaystyle b_{1}(M)=0}$
• The virtual hook conjecture , proven by Ian Agol , says that every irreducible 3-manifold is finitely overlaid by a hook manifold .

Hierarchies

For a hook manifold with an incompressible surface there is a sequence ${\displaystyle M}$ ${\displaystyle F\subset M}$

${\displaystyle (M_{0},F_{0}),(M_{1},F_{1}),\ldots ,(M_{k},F_{k}),(M_{k+1},F_{k+1})}$,

so that , from is created by cutting along and is a union of disjoint 3-dimensional full spheres. ${\displaystyle (M,F)=(M_{0},F_{0})}$ ${\displaystyle (M_{i+1},F_{i+1})}$ ${\displaystyle (M_{i},F_{i})}$ ${\displaystyle F_{i}}$ ${\displaystyle M_{k+1}}$

This property makes it possible to carry out proofs for hook manifolds as induction proofs over the length of a hook hierarchy, whereby the induction start consists in checking the assertion for 3-dimensional solid spheres. In this way Waldhausen's rigidity theorem for hook manifolds and Thurston's geometrization conjecture for hook manifolds were proven.

Waldhausen's Theorem of Rigidity

Theorem (Waldhausen): Let be a closed hook manifold. Then every homotopy equivalence is homotopic to a homeomorphism . This applies accordingly to hooked manifolds with a border, if one assumes that the homotopy equivalence on the border is already a homeomorphism. ${\displaystyle M}$ ${\displaystyle \partial M}$

Algorithmic aspects

There is an algorithm that decides whether two hook manifolds are homeomorphic. This algorithm, known under the name "Recognition Theorem", is theoretical in nature. In particular, one has an algorithmic classification of hook manifolds and thus (because of the Gordon-Luecke theorem ) also an algorithmic classification of knots and links . (Gordon-Luecke's theorem does not apply to links with several components, but these are clearly determined by the complement and its meridians.)

There is also an algorithm implemented on the computer to decide whether an irreducible 3-manifold is a hook.

Higher-dimensional hook manifold

A fringe pattern (ger .: boundary pattern) is a finite set of compact coherent -dimensional submanifolds of the border ( "facets"), so for the average of each of these submanifolds one -dimensional submanifold or empty. The boundary pattern is called complete when the union of these submanifolds is complete , and useful when ${\displaystyle (n-1)}$ ${\displaystyle \partial M}$ ${\displaystyle 1\leq k\leq n+1}$ ${\displaystyle k}$ ${\displaystyle (n-k)}$ ${\displaystyle \partial M}$

• every zero-homotopic mapping of into a facet is already zero-homotopic in the facet ${\displaystyle M}$ ${\displaystyle S^{1}}$
• each zero-homotopic map consisting of two intervals mapped into a facet from in to a map from in bordered, which intersects the average of the two facets in a single interval ${\displaystyle S^{1}}$ ${\displaystyle \partial M}$ ${\displaystyle D^{2}}$ ${\displaystyle \partial M}$
• each zero-homotopic mapping consisting of three intervals each mapped into a facet from in to a mapping from in bordered, which intersects the edge of the three facets in a single tripod ${\displaystyle S^{1}}$ ${\displaystyle \partial M}$ ${\displaystyle D^{2}}$ ${\displaystyle \partial M}$

${\displaystyle n}$-dimensional hook cells are certain -manifolds with border patterns, which are recursively defined as follows. A -dimensional hook cell is a corner ( ) with the edges as a border pattern. A -dimensional hook-cell is a manifold with a complete and useful border pattern, the elements of which are -dimensional hook-cells. ${\displaystyle n}$ ${\displaystyle 2}$ ${\displaystyle k}$ ${\displaystyle k\geq 4}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$

A -dimensional manifold is a hook manifold if it is a sequence ${\displaystyle n}$ ${\displaystyle M}$

${\displaystyle (M_{0},F_{0}),(M_{1},F_{1}),\ldots ,(M_{k},F_{k}),(M_{k+1},F_{k+1})}$

of manifolds with complete and useful boundary patterns as well as -dimensional submanifolds , so that out is created by slicing along and the boundary pattern of is created by that of , and so that and is a disjoint union of -dimensional hook cells. ${\displaystyle M_{i}}$ ${\displaystyle (n-1)}$ ${\displaystyle F_{i}\subset M_{i}}$ ${\displaystyle M_{i+1}}$ ${\displaystyle M_{i}}$ ${\displaystyle F_{i}}$ ${\displaystyle M_{i+1}}$ ${\displaystyle M_{i}}$ ${\displaystyle M_{0}=M}$ ${\displaystyle M_{k+1}}$ ${\displaystyle n}$

Examples

• Areas of non-positive Euler characteristics with the edge components as edge patterns.
• A -dimensional hook-manifold with an incompressible boundary and its components as boundary pattern is a hook-manifold in the sense of this definition. ${\displaystyle 3}$
• A -manifold of shape , where is a -dimensional closed hook manifold, with the edge components as the edge pattern. ${\displaystyle 4}$ ${\displaystyle N\times \left[0,1\right]}$ ${\displaystyle N}$ ${\displaystyle 3}$
• A -manifold of shape , where a surface is of non-positive Euler characteristic, with an edge pattern consisting of four copies of . ${\displaystyle 4}$ ${\displaystyle S\times \left[0,1\right]^{2}}$ ${\displaystyle S}$ ${\displaystyle S\times \left[0,1\right]}$

properties

• ${\displaystyle n}$-dimensional hooked manifolds are aspherical, their universal superposition is homeomorphic to the . ${\displaystyle \mathbb {R} ^{n}}$
• The word problem for the fundamental groups of hook manifolds is solvable.

literature

• W. Haken: Theory of normal surfaces I. In: Acta Math. 105, 1961, pp. 245-375.
• F. Waldhausen: On irreducible 3-manifolds which are sufficiently large. In: Ann. of Math. 87, 1968, pp. 56-88.
• W. Jaco: Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43rd American Mathematical Society, Providence, RI, 1980. ISBN 0-8218-1693-4
• B. Foozwell, H. Rubinstein: Introduction to the theory of Haken n-manifolds. In: Topology and geometry in dimension three. (= Contemp. Math. 560). Amer. Math. Soc., Providence, RI, 2011, ISBN 978-0-8218-5295-8 , pp. 71-84.

Web links

• William Jaco : Haken manifold (Encyclopedia of Mathematics)
• Johannson's characteristic submanifold theory , Chapter 2 in: Canary, McCullough, Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups

Individual evidence

1. ^ William Thurston : Geometry and topology of three manifolds . Chapter 4: Hyperbolic Dehn surgery (pdf)
2. ↑ Sergei Matveev: Algorithmic topology and classification of 3-manifolds. (= Algorithms and Computation in Mathematics. 9). 2nd Edition. Springer, Berlin 2007, ISBN 978-3-540-45898-2 , chapter 6.
3. ^ Klaus Johannson : Homotopy equivalences of 3 manifolds with boundaries. (= Lecture Notes in Mathematics. 761). Springer, Berlin 1979, ISBN 3-540-09714-7 .