3-manifold

In mathematics, spaces are called 3-manifold or 3-dimensional manifold that look locally like the 3-dimensional Euclidean space .

Examples

Euclidean space

→ Main article : Euclidean space

Euclidean space is the simplest example of a 3-manifold. It is non-compact and simply connected . Every 3-manifold is locally homeomorphic to the . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

The Euclidean metric on the is a flat metric , i.e. its curvature of intersection is constantly zero. But there are numerous other Riemannian metrics on the . In particular, the space is homeomorphic to the hyperbolic space , the curvature of which is constant −1. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

The Hopf fiber from over : Points on the 2-sphere have the same color as the fiber of the 3-sphere above them. ${\displaystyle S^{3}=\mathbb {R} ^{3}\cup \left\{\infty \right\}}$ ${\displaystyle S^{2}}$

3 sphere

→ Main article : 3-sphere

The 3-dimensional sphere is compact and simply connected. The Poincaré conjecture , proved by Perelman , states that it is the only simply connected, closed manifold . So you are the simplest closed 3-manifold.

The embedding as a unitary sphere in the equips it with a Riemannian metric, the cutting curvature of which is constant +1. ${\displaystyle \mathbb {R} ^{4}}$

SU (2)

→ Main article : SU (2)

The Lie group SU (2) is diffeomorphic to the 3-sphere.

${\displaystyle SU(2)}$is a twofold superposition of and especially isomorphic to the spin group , which is therefore also diffeomorphic to the spin group . ${\displaystyle SO(3)}$ ${\displaystyle Spin(3)}$ ${\displaystyle S^{3}}$

Whitehead manifold

→ Main article : Whitehead manifold

The Whitehead manifold is a simply connected, non-compact 3-manifold that is not homeomorphic because it is not "simply connected at infinity". Whitehead discovered it as a counterexample to an analogue of the Poincaré conjecture for non-compact manifolds. ${\displaystyle \mathbb {R} ^{3}}$

3 torus

→ Main article : “General definition of a torus” in the article Torus

The 3-dimensional torus is obtained by identifying the opposite side faces of a cube , or as the product space of three circles .

Its fundamental group is the free Abelian group , its universal superposition is the . ${\displaystyle \mathbb {Z} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

The 3-torus carries flat metrics; H. Riemannian metrics of the sectional curvature constant zero. Each such metric is obtained by realizing the 3-torus as for a lattice . The module space of such grids is that is the module space of flat metrics . ${\displaystyle \mathbb {R} ^{3}/L}$ ${\displaystyle L\subset \mathbb {R} ^{3}}$ ${\displaystyle SL(3,\mathbb {Z} )\backslash SL(3,\mathbb {R} )}$ ${\displaystyle SL(3,\mathbb {Z} )\backslash SL(3,\mathbb {R} )/SO(3)}$

Projective space

→ Main article : Projective space

The projective space is the quotient space of the unit sphere with regard to the identification for all . It therefore has the fundamental group , universal superposition , and it is a spherical manifold, i. H. it carries a Riemannian metric of the cutting curvature constant 1. ${\displaystyle \mathbb {R} P^{3}}$ ${\displaystyle x\sim -x}$ ${\displaystyle x\in S^{3}\subset \mathbb {R} ^{4}}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle S^{3}}$

The projective linear group acts on . ${\displaystyle PGL(3,\mathbb {R} )}$ ${\displaystyle \mathbb {R} P^{3}}$

SO (3)

→ Main article : Rotation group # Three-dimensional rotations

The Lie group is diffeomorphic to the . ${\displaystyle SO(3)}$ ${\displaystyle \mathbb {R} P^{3}}$

Poincaré homology sphere

The Poincaré homology sphere is a spherical 3-manifold whose fundamental group has the order 120. Their homology groups are isomorphic to those of . ${\displaystyle S^{3}}$

They are constructed as quotients , the archetype of group A _{5 being} the orientation- preserving symmetries of the regular dodecahedron under the double superposition . ${\displaystyle S^{3}/\Gamma }$ ${\displaystyle \Gamma \subset SU(2)=S^{3}}$ ${\displaystyle \subset SO(3)}$ ${\displaystyle SU(2)\to SO(3)}$

Figure eight knot

Whitehead Entanglement

Weeks manifold

The Weeks manifold is the hyperbolic 3-manifold of the smallest hyperbolic volume . It is obtained by (5,1) and (5,2) stretching surgery on the two components of the Whitehead link.

Gieseking manifold

The Gieseking manifold is the manifold of the smallest hyperbolic volume among the non-compact, hyperbolic 3-manifolds. It arises from a regular ideal tetrahedron through a suitable identification of two pairs of side faces. In particular, it has the volume of a regular ideal tetrahedron, i.e. the Gieseking constant 1.0149 ...

Figure eight complement

→ Main article : figure eight knot complement

The complement of the figure-eight knot in the 3-sphere, together with its sister manifold constructed by (5,1) -stretching surgery on one of the two components of the Whitehead linkage, is the manifold of the smallest hyperbolic volumes among the orientable, non-compact, hyperbolic 3- Manifolds. It is a 2-fold superposition of the Gieseking manifold, so its volume is twice the Gieseking constant.

It is a bundle of fibers above the circle, the fiber of which is a dotted torus and the monodromy of which is Arnold's cat image .

Complement of the Whitehead Link

The complement of the Whitehead link is a hyperbolic 3-manifold with two tips . A fundamental domain in hyperbolic space is the regular ideal octahedron . The hyperbolic volume of the complement of the Whitehead link is therefore 3.663862377 ..., the volume of the regular ideal octahedron. The complements of the Whitehead link and its "sister", the (-2,3,8) - pretzel link , are the two orientable, hyperbolic 3-manifolds of the smallest volume, the edge of which consists of at least two connected components.

Classes of 3-Manifolds

Spherical 3-manifold

A spherical 3-manifold is a Riemannian manifold with constant intersection curvature . It is equivalent in shape , being the 3-sphere and a discrete subgroup of its isometric group . Then you have . ${\displaystyle M}$ ${\displaystyle +1}$ ${\displaystyle \Gamma \backslash S^{3}}$ ${\displaystyle S^{3}}$ ${\displaystyle \Gamma \subset Isom(\mathbb {S} ^{3})}$ ${\displaystyle \pi _{1}M=\Gamma }$

Because of , the spherical 3-manifolds correspond one-to-one to the finite subgroups of . ${\displaystyle Isom(S^{3})=SO(4)}$ ${\displaystyle SO(4)}$

Lens room

→ Main article : lens room

Lens spaces are spherical manifolds where there is a cyclic group . ${\displaystyle M}$ ${\displaystyle \pi _{1}M}$

In contrast to hook manifolds, for lens spaces the homeomorphism type and even the homotopy equivalence class are not yet determined by the fundamental group . Reidemeister proved by means of the subsequently named after him Reidemeister torsion that the homotopieäquivalenten lens spaces and are not homeomorphic. ${\displaystyle L(7,1)}$ ${\displaystyle L(7,2)}$

Seifert fiber

→ Main article : Seifert fiber

A Seifert fiber is a 3-manifold that can be broken down homeomorphically into fibers , so that each fiber either has an environment homeomorphic to (regular fiber) or an environment homeomorphic to the mapping torus of the rotation of the circular disk around the angle (singular fiber of type ) owns. ${\displaystyle S^{1}}$ ${\displaystyle S^{1}\times D^{2}}$ ${\displaystyle 360^{\circ }\cdot q/p}$ ${\displaystyle (p,q)}$

Seifert fibers are the simplest pieces in the JSJ decomposition of 3-manifolds.

I bundle

I-bundles are 3-manifolds with a border, which are a fiber bundle with fibers homeomorphic to the interval over a compact surface (possibly with a border). They occur in the JSJ decomposition of manifolds with a non-empty margin. ${\displaystyle I=\left[0,1\right]}$

Graph manifold

Waldhausen originally defined graph manifolds as 3-manifolds that can be broken down into bundles of circles by cutting along disjoint, embedded tori . An equivalent condition is that they are connected sums of 3-manifolds which are either Solv-manifolds or whose JSJ decomposition only contains Seifert fibers.

Hyperbolic 3-manifold

→ Main article : Hyperbolic manifold

A hyperbolic 3-manifold is a complete Riemannian manifold with constant intersection curvature . It is equivalent in shape , being 3-dimensional hyperbolic space and a discrete subgroup of the group of isometries of hyperbolic space. Then you have . ${\displaystyle M}$ ${\displaystyle -1}$ ${\displaystyle \Gamma \backslash \mathbb {H} ^{3}}$ ${\displaystyle \mathbb {H} ^{3}}$ ${\displaystyle \Gamma \subset Isom(\mathbb {H} ^{3})}$ ${\displaystyle \pi _{1}M=\Gamma }$

Because of this , the orientable, hyperbolic 3-manifolds correspond one-to-one to the conjugation classes of discrete subgroups of . ${\displaystyle Isom^{+}(H^{3})=PSL(2,\mathbb {C} )}$ ${\displaystyle PSL(2,\mathbb {C} )}$

The hyperbolic theorem says that a closed 3-manifold is hyperbolic if and only if it is irreducible, has an infinite fundamental group and no subgroup that is too isomorphic in its fundamental group. ${\displaystyle \mathbb {Z} ^{2}}$

Hook manifold

→ Main article : Hook manifold

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

Hook manifolds have hierarchies of incompressible surfaces through which they can be broken down into a union of disjoint 3-dimensional full spheres. This enables proofs for hook manifolds to be carried out as induction proofs over the length of a hook hierarchy.

Fibred 3-manifold

→ Main article : Image torus

A fibered 3-manifold is a 3-manifold of shape

${\displaystyle M_{f}:=\Sigma \times \left[0,1\right]/\sim ,\quad (x,0)\sim (f(x),1)\ \forall x\in \Sigma }$

for a surface and a homeomorphism . ${\displaystyle \Sigma }$ ${\displaystyle f\colon \Sigma \to \Sigma }$

The homeomoorphy type of depends only on the mapping class of . A 3-manifold can fiber in different ways. ${\displaystyle M_{f}}$ ${\displaystyle f}$

From the geometry of 3-manifolds it follows for surfaces of gender : ${\displaystyle \Sigma }$ ${\displaystyle \geq 2}$

• ${\displaystyle M_{f}}$is a Seifert fiber if and only if the mapping class of is periodic , ${\displaystyle f}$
• ${\displaystyle M_{f}}$has a nontrivial JSJ-decomposition if and only if the mapping class of is reducible . ${\displaystyle f}$
• ${\displaystyle M_{f}}$is hyperbolic if and only if the mapping class is from pseudo-Anosovian . ${\displaystyle f}$

If the fiber is a surface of genus is thus a torus, is obtained in the case that Anosovsch a is sol-structure on . ${\displaystyle 1}$ ${\displaystyle f}$ ${\displaystyle M_{f}}$

Node complement

→ Main article : Node complement

A node complement is the space remaining after removing a node from the 3-sphere.

Two nodal complements are homeomorphic if and only if the nodes are equivalent. The corresponding statement for entanglements does not apply.

From the geometrization of 3-manifolds it follows:

• a knot complement is a Seifert fiber if and only if the knot is a torus knot ,
• a node complement has a nontrivial JSJ decomposition if and only if the node is a satellite node ,
• a nodal complement is a hyperbolic manifold if and only if neither of the other two cases applies.

Construction principles

Heegaard decomposition

→ Main article : Heegaard decomposition

A Heegaard decomposition of a closed 3-dimensional manifold consists of two handle bodies and and a homeomorphism , so that results from and through gluing means . From the Morse theory it follows that every closed orientable 3-manifold has a Heegaard decomposition. ${\displaystyle M}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle f:\partial H_{1}\rightarrow \partial H_{2}}$ ${\displaystyle M}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle f}$

Stretching surgery

→ Main article : Stretching surgery

Stretching surgery is a process for the construction of 3-dimensional manifolds by drilling a knot out of the 3-dimensional sphere and gluing it back in another way.

Every closed, orientable, connected 3-manifold can be constructed by stretching surgery on a loop in the 3-sphere. ${\displaystyle L\subset S^{3}}$

Triangulation

→ Main article : Triangulation (topology)

Pachner train.

A triangulation of a 3-manifold is given by a 3-dimensional simplicial complex and a homeomorphism the geometrical implementation on . ${\displaystyle M}$ ${\displaystyle K}$ ${\displaystyle h\colon \vert K\vert \rightarrow X}$ ${\displaystyle \vert K\vert }$ ${\displaystyle M}$

Moise proved in 1952 that all 3-manifolds can be triangulated and that for 3-manifolds the main conjecture applies: every two triangulations of the same manifold have a common subdivision. In particular, 3-manifolds have a unique PL structure .

Any two different triangulations of the same manifold can be converted into one another by a sequence of Pachner trains .

Invariants

Fundamental group

The fundamental group is an important invariant of closed 3-manifolds. Non-spherical geometric 3-manifolds and irreducible 3-manifolds with nontrivial JSJ decomposition are already clearly defined by their fundamental group.

The rank of the fundamental group is denoted by. From the Poincaré conjecture it follows that the 3-sphere is the only closed 3-manifold with . It follows from Grushko's theorem ${\displaystyle \pi _{1}M}$ ${\displaystyle rk(M)}$ ${\displaystyle rk(M)=0}$

${\displaystyle rk(M_{1}\sharp M_{2})=rk(M_{1})+rk(M_{2})}$.

Homology groups

The homology groups of a closed, orientable 3-manifold are already clearly defined by their fundamental group. It is namely the Abelization of the fundamental group, because of Poincaré duality its dual (hence the quotient from to its torsion subgroup ), as well as . ${\displaystyle M}$ ${\displaystyle H_{1}(M)}$ ${\displaystyle H_{2}(M)}$ ${\displaystyle H_{1}(M)}$ ${\displaystyle H_{0}(M)=H_{3}(M)=\mathbb {Z} }$

Thurston standard

The Thurston norm is a seminorm on the second homology group of an oriented 3-manifold. It measures the complexity of the embedded surfaces representing the homology class. ${\displaystyle H_{2}(M)}$

Embedded areas that minimize the Thurston norm in their homology class are leaves of tight foliage.

Hyperbolic volume

→ Main article : Hyperbolic volume

Hyperbolic volume is a topological invariant because, according to Mostow-Prasad's rigidity, there can at most be one hyperbolic metric of finite volume on a manifold of dimension . A generalization to arbitrary (not necessarily hyperbolic) manifolds is the simplicial volume , which in the case of 3-manifolds gives the sum of the volumes of the hyperbolic pieces in the JSJ decomposition (multiplied by the inverse of Gieseking's constant ). ${\displaystyle {\displaystyle \geq 3}}$

The volume conjecture establishes a connection between hyperbolic volume and quantum invariants of nodes, which has so far only been proven in a few cases.

The hyperbolic volumes of 3-manifolds form a well-ordered subset of the real numbers, i.e. H. every family of hyperbolic 3-manifolds has a least-volume element. There are only finitely many 3-manifolds with the same volume. Gabai-Meyerhoff-Milley developed mom technology to create complete lists of hyperbolic manifolds of small volume.

Chern-Simons invariant

→ Main article : “Theorem of Yoshida” in the article Chern-Simons functional

Let be a closed, orientable, hyperbolic 3-manifold and its monodrome representation, then applies to the associated flat bundle ${\displaystyle M}$ ${\displaystyle \rho :\pi _{1}M\rightarrow PSL(2,\mathbb {C} )}$ ${\displaystyle E_{\rho }}$

${\displaystyle CS(E_{\rho })=cs(M)+{\frac {i}{2\pi ^{2}}}vol(M)\in \mathbb {C} /\mathbb {Z} }$,

where the Riemann Chern-Simons invariant of the Levi-Civita relationship denotes. ${\displaystyle cs(M)}$

The right side of this equation is also known as the complex volume .

The picture of the fundamental class below the representation defines a homology class ${\displaystyle \rho }$

${\displaystyle (B\rho )_{*}\left[M\right]\in H_{3}(PSL(2,\mathbb {C} )^{\delta };\mathbb {Z} )\simeq {\hat {B}}(\mathbb {C} )}$

in the expanded Bloch group and the Rogers dilogarithm

${\displaystyle R:{\hat {B}}(\mathbb {C} )\rightarrow \mathbb {C} /2\pi ^{2}\mathbb {Z} }$

maps to . This provides an explicit formula for the Chern-Simons invariant and an alternative proof of Yoshida's theorem. ${\displaystyle (B\rho )_{*}\left[M\right]}$ ${\displaystyle CS(E_{\rho })}$

Casson invariant

The Casson invariant is an invariant of 3-dimensional spheres of homology. For a Heegaard decomposition it is the number of cuts of the - character varieties and in ${\displaystyle M=H_{1}\cup _{S_{g}}H_{2}}$ ${\displaystyle {\tfrac {(-1)^{g}}{2}}}$ ${\displaystyle SU(2)}$ ${\displaystyle {\mathcal {R}}(H_{1})}$ ${\displaystyle {\mathcal {R}}(H_{2})}$ ${\displaystyle {\mathcal {R}}(S_{g})}$

Heegaard gender

The Heegaard gender of a compact, orientable 3-manifold is the minimal gender of the Heegaard surface in a Heegaard decomposition of . It always applies . An open conjecture says that for hyperbolic manifolds . The assumption is generally wrong for Seifert fibers. ${\displaystyle g(M)}$ ${\displaystyle M}$ ${\displaystyle g(M)\geq rk(M)}$ ${\displaystyle g(M)=rk(M)}$

Heegaard-Floer homology

→ Main article : Heegaard-Floer homology

Heegaard-Floer homology is an invariant of a closed spin ^c -3 manifold . It is constructed using the Heegaard decomposition of Lagrange-Floer homology. Several homology groups are obtained, which are related to one another through exact sequences. ${\displaystyle M}$ ${\displaystyle M}$

Heegaard-Floer homology can be used to distinguish the unknot from all non-trivial knots.

Reidemeister Twist

→ Main article : Reidemeister twist

The Reidemeister torsion can be used to distinguish the homotopy-equivalent lens spaces for which other invariants of the algebraic topology match.

L ² invariants

Because the fundamental groups of 3-manifolds are residual finite , there is a descending sequence with and . Then, according to Lück's approximation theorem, the L ² -Betti numbers can be passed through ${\displaystyle \pi _{1}M}$ ${\displaystyle \Gamma _{n}\vartriangleleft \pi _{1}M}$ ${\displaystyle \left[\pi _{1}M\colon \Gamma _{n}\right]<\infty }$ ${\displaystyle \cap _{n}\Gamma _{n}=0}$

${\displaystyle b_{i}^{(2)}(M)=\lim _{n}{\frac {b_{i}(M)}{\left[\pi _{1}M\colon \Gamma _{n}\right]}}}$

to calculate. The L ² torsion of 3-manifolds is proportional to the simplicial volume.

Turaev Viro Invariants

Turaev-Viro-invariants are invariants of closed 3-manifolds defined by means of state sums.

Node invariants

Because, according to Gordon-Luecke's theorem, node complements are homeomorphic if and only if the nodes are equivalent, node invariants such as quantum invariants and the Alexander polynomial are also topological invariants of node complements.

Structures on 3-manifolds

(G, X) structure

→ Main article : (G, X) structure

A manifold has a -structure for a transitive G-space , if it can be covered locally homeomorphically by open sets (“maps”) , so that the coordinate transitions are restrictions of elements . ${\displaystyle (G,X)}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle G}$

A model geometry is a differentiable manifold with a more differentiated effect of a Lie group that satisfies the following conditions: ${\displaystyle X}$ ${\displaystyle G}$

• ${\displaystyle X}$ is connected and simply connected
• ${\displaystyle G}$acts transitively with compact stabilizers (in particular there is an -invariant Riemannian metric) ${\displaystyle X}$ ${\displaystyle G}$
• ${\displaystyle G}$is maximal among groups that act through diffeomorphisms with compact stabilizers ${\displaystyle X}$
• there is at least one compact manifold. ${\displaystyle (G,X)}$

From the last condition it follows in particular that it must be unimodular . There are numerous pairs that satisfy all but the last, for example , the Lie group of affine maps of the Euclidean plane. ${\displaystyle G}$ ${\displaystyle (G,X)}$ ${\displaystyle G=X=Aff(\mathbb {R} ^{2})}$

Thurston has proven that there are exactly eight 3-dimensional model geometries:

• the Euclidean space , ${\displaystyle \mathbb {R} ^{3}}$
• the three-dimensional sphere (surface of a four-dimensional sphere), ${\displaystyle S^{3}}$
• the hyperbolic space , ${\displaystyle \mathbb {H} ^{3}}$
• the product of 2-sphere and straight line , ${\displaystyle S^{2}\times \mathbb {R} }$
• the product of the hyperbolic plane and the straight line , ${\displaystyle H^{2}\times \mathbb {R} }$
• ${\displaystyle {\widetilde {\mathrm {SL} }}(2,\mathbb {R} )}$, the universal superposition of the special linear group ${\displaystyle \mathrm {SL} (2,\mathbb {R} )}$
• the Heisenberg Group ${\displaystyle Nil}$
• the 3-dimensional resolvable Lie group . ${\displaystyle Sol}$

Cross section through a reeb foliage.

Foliage

→ Main article : Foliage

Every 3-manifold has leaves of codimension 1, but in general these leaves have Reeb components .

Gabai has proven that every 3-manifold with has a tight foliage . Tight leaves have no Reeb components. ${\displaystyle H_{2}(M,\partial M)\not =0}$

lamination

→ Main article : Lamination (math)

A lamination of a manifold is a foliation of a closed subset of . ${\displaystyle M}$ ${\displaystyle M}$

In the theory of 3-manifolds, essential laminations are of particular importance.

Special cases of essential lamination are incompressible surfaces and tight leaves.

The standard contact structure of the . ${\displaystyle \mathbb {R} ^{3}}$

Contact structure

→ Main article : Contact geometry

In 3-dimensional contact geometry there is a dichotomy between tight and over-twisted contact structures.

Eliashberg and Thurston have proven that every foliation of a 3-manifold (with the exception of the product foliation of ) can be approximated by contact structures and, in particular, tight foliations can be approximated by tight contact structures. Hence the existence of tight contact structures on 3-manifolds follows from Gabai's theorem . ${\displaystyle S^{2}\times S^{1}}$ ${\displaystyle H_{2}(M,\partial M)\not =0}$

Basic results

Moise's theorem

Moise's theorem says that every 3-manifold has a unique PL structure and a unique differential structure.

Prime decomposition

→ Main article : Prime decomposition (topology)

The prime decomposition of a closed, connected -dimensional manifold is a decomposition as a connected sum of finitely many prime manifolds, i.e. ${\displaystyle 3}$ ${\displaystyle M}$

${\displaystyle M=M_{1}\sharp \ldots \sharp M_{k}}$

with prime manifolds (the prime components ). ${\displaystyle M_{1},\ldots ,M_{k}}$

The existence of the prime decomposition for 3-manifolds was proved in 1924 by Kneser, its uniqueness in 1962 by Milnor.

JSJ decomposition

→ Main article : JSJ decomposition

A theorem by Jaco-Shalen and Johannson states that every irreducible, closed 3-dimensional manifold has a Seifert-fibered submanifold with atoroidal complement that is unique except for isotopically (not necessarily connected). This is also referred to as the characteristic submanifold.

The JSJ decomposition is an important prerequisite for the geometrization of 3-manifolds. Every Seifert-fibered manifold can be geometrized and every atoroidal irreducible 3-manifold carries a hyperbolic metric.

For manifolds with a boundary one also has a JSJ decomposition, here the characteristic submanifold consists not only of Seifert fibers, but also of I-bundles.

Dehn's lemma

→ Main article : Dehn's lemma

Dehn's lemma states that an embedded circle bounding an immersed disc in a 3-manifold also bounds an embedded disc.

Set of spheres

→ Main article : Theorem of spheres (topology)

The theorem of spheres says that in a 3-manifold with a nontrivial second homotopy group there must always be embedded, non-null-homotopic 2-spheres.

The finiteness of Kneser and Haken

The finiteness theorem proved by Wolfgang Haken says that there is an integer for a compact, irreducible 3-manifold , so that for every set of disjoint, embedded, two-sided, incompressible surfaces one of the components of a product must be. ${\displaystyle M}$ ${\displaystyle c(M)}$ ${\displaystyle k>c(M)}$ ${\displaystyle \left\{S_{1},\ldots ,S_{k}\right\}}$ ${\displaystyle \textstyle M\setminus \bigcup _{i=1}^{k}S_{i}}$ ${\displaystyle S_{i}\times \left[0,1\right]}$

The corresponding theorem for embedded 2-spheres had already been proven by Kneser and was the main step in the proof of the existence of the prime decomposition.

Torus theorem

→ Main article : Torus theorem

Let it be an orientable irreducible 3-manifold whose fundamental group contains a subgroup isomorphic to . Then there is either a Seifert fiber or there is an embedded incompressible torus . ${\displaystyle M}$ ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$ ${\displaystyle M}$ ${\displaystyle T^{2}\subset M}$

Sentence about the compact core

Every 3-manifold with a finitely generated fundamental group has a compact kernel, i.e. H. a compact submanifold whose inclusion in a homotopy equivalence is. ${\displaystyle M}$ ${\displaystyle M}$

Lickorish-Wallace theorem

→ Main article : Lickorish-Wallace theorem

Every closed, orientable, connected 3-manifold can be constructed by stretching surgery on a link in the 3-sphere. One can even achieve that all components of are untied and that all are coefficients . ${\displaystyle L\subset S^{3}}$ ${\displaystyle L}$ ${\displaystyle -{\tfrac {b_{i}}{d_{i}}}=\pm 1}$

Poincaré conjecture

→ Main article : Poincaré conjecture

The Poincaré conjecture proved by Perelman says that every compact, simply connected 3-manifold is homeomorphic to . ${\displaystyle S^{3}}$

Hyperbolization

The Thurston conjecture, proven by Grigori Perelman, states that every atoroidal irreducible 3-manifold has a hyperbolic metric.

Geometrization

→ Main article : Geometrization of 3-manifolds

The goal of the geometrization is to find a characteristic geometric structure on each of these building blocks after breaking down a 3-manifold into basic building blocks. The conjecture made by Thurston that this is always possible represents a generalization of the Poincaré conjecture and was proven by Grigori Perelman with his work on the Ricci River.

Precisely the geometrization says that the pieces of the JSJ decomposition of a compact 3-manifold have a -structure. ${\displaystyle (G,X)}$

Seifert fiber space conjecture

→ Main article : Seifert fiber space conjecture

The Seifert fiber space conjecture proved by Casson-Jungreis and Gabai says that a 3-manifold is a Seifert fiber if and only if the center of its fundamental group is isomorphic to the group of whole numbers . ${\displaystyle \mathbb {Z} }$

Waldhausen's Theorem of Rigidity

→ Main article : Waldhausen's theorem of rigidity

Waldhausen's theorem of rigidity states that every homotopy equivalence between hook manifolds is homotopic to a homeomorphism. In particular, hook manifolds are uniquely determined by their fundamental groups.

Waldhausen conjecture

The Waldhausen conjecture proved by Tao Li says that a closed, orientable, irreducible, atoroidal 3-manifold up to isotopia only has a finite number of Heegaard decompositions with Heegaard surfaces of a given gender . ${\displaystyle g}$

Smith conjecture

The Smith conjecture said that diffeomorphisms of finite order have an unknotted set of fixed points . It was proved in the 80s with the help of the geometrization of 3-manifolds. ${\displaystyle f\colon S^{3}\to S^{3}}$

Sentence about cyclic surgery

Let be a connected, compact, orientable, irreducible 3-manifold which is not a Seifert fiber and whose edge is a torus. The sentence about cyclic surgery says: if two different stretch fillings lead to manifolds with cyclic fundamental groups, then the number of intersections of the edges of the meridians of the two filled full gates is at most 1. ${\displaystyle M}$

Hyperbolic Stretching Surgery

Thurston's theorem on hyperbolic stretching surgery states that almost all manifolds produced by stretching surgery on a given hyperbolic node are also hyperbolic. ${\displaystyle K}$

Tameness phrase

→ Main article : Tameness Theorem

The Marden conjecture, proven by Agol and Calegari-Gabai, states that every complete , 3-dimensional hyperbolic manifold with a finitely generated fundamental group is topologically tame, i.e. homeomorphic to the interior of a compact manifold.

Lemma of Margulis

→ Main article : Lemma of Margulis

Margulis' lemma describes the thin parts of a hyperbolic manifold.

In particular, one obtains that a hyperbolic 3-manifold has finite hyperbolic volume if and only if it is the interior of a compact manifold whose boundary consists of incompressible tori or is empty.

Mostow's law of rigidity

→ Main article : Mostow rigidity

Hyperbolic metrics of finite volume are unique on a 3-manifold, if they exist, except for isometry. Equivalently there is only a discrete embedding of the fundamental group in the isometric group of the 3-dimensional hyperbolic space except for conjugation .

In particular, geometrically defined invariants such as volume, Chern-Simons invariant and length spectrum are also topological invariants of hyperbolic manifolds of finite volume.

Geometrically finite Klein groups

→ Main article : Geometrically finite group # isometric groups of 3-dimensional hyperbolic space (Klein groups)

Let be a hyperbolic 3-manifold of infinite volume. The Klein group is geometrically finite if it satisfies one of the following equivalent conditions: ${\displaystyle M=\Gamma \backslash \mathbb {H} ^{3}}$ ${\displaystyle \Gamma \subset Isom(\mathbb {H} ^{3})}$

• there is a fundamental polyhedron with a finite number of faces
• the Dirichlet area has a finite number of faces for everyone ${\displaystyle x\in \mathbb {H} ^{3}}$
• the convex kernel of has finite volume. ${\displaystyle C(M)}$ ${\displaystyle M=\Gamma \backslash \mathbb {H} ^{3}}$

Geometrically finite hyperbolic metrics on a given 3-manifold are uniquely determined by their conformal boundaries (i.e. the quotients of the regions of discontinuity in the sphere at infinity ).

Sentence about end laminations

An end of a hyperbolic 3-manifold is called geometrically finite if it has a neighborhood that is disjoint from the convex kernel . Otherwise the end is called geometrically infinite . If one end of a hyperbolic 3-manifold is geometrically infinite, then for every neighborhood of there is a closed geodesic with . For a geometrically infinite end of the shape , the end lamination is defined as the lamination of the surface , which is obtained as the limit value of a (each) sequence of geodesics that ultimately leave each compact subset . ${\displaystyle M}$ ${\displaystyle C(M)}$ ${\displaystyle e}$ ${\displaystyle M}$ ${\displaystyle U}$ ${\displaystyle e}$ ${\displaystyle \gamma \subset M}$ ${\displaystyle \gamma \cap U\not =\emptyset }$ ${\displaystyle S\times (0,\infty )}$ ${\displaystyle S}$ ${\displaystyle \gamma _{n}\subset M}$

The theorem on end lamination , proven by Jeffrey Brock, Richard Canary and Yair Minsky, states that geometrically infinite ends are uniquely determined by their end lamination.

Thurston-Bonahon's theorem

→ Main article : Thurston-Bonahon's theorem

Thurston-Bonahon's theorem states that a closed surface in a 3-hyperbolic manifold is either quasi-geodesic or a virtual fiber .

Theorem about groups of surfaces

The "surface subgroup conjecture", which goes back to Waldhausen and was proven by Kahn-Markovic, says that the fundamental group of an irreducible, non-spherical, closed 3-manifold contains a subgroup isomorphic to a surface group .

Virtual hook manifolds

The "virtual hook conjecture" (VHC) proved by Ian Agol says that every 3-manifold has a finite cover , which is a hook-manifold.

Virtual fibers

The "virtual fibered conjecture" (VFC) proved by Ian Agol says that every closed 3-manifold has a finite overlay that is a bundle of surfaces over the circle.

literature

• Herbert Seifert , William Threlfall : Textbook of Topology. Teubner 1934.
• John Hempel : 3-Manifolds. Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1976.
• William Jaco : Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43rd American Mathematical Society, Providence, RI, 1980. ISBN 0-8218-1693-4
• Riccardo Benedetti , Carlo Petronio : Lectures on hyperbolic geometry. University text. Springer-Verlag, Berlin, 1992. ISBN 3-540-55534-X
• William Thurston : Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. ISBN 0-691-08304-5
• Michael Kapovich : Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhauser Boston, Inc., Boston, MA, 2001. ISBN 0-8176-3904-7
• Nikolai Saweliew : Invariants for homology 3-spheres. Encyclopaedia of Mathematical Sciences, 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002. ISBN 3-540-43796-7
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