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ATOROIDAL

Atoroidal

In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential

Atoroidal

Hyperbolization theorem

Theorem in geometry

geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston

Hyperbolization theorem

Hyperbolization_theorem

JSJ decomposition

Process in mathematics of decomposing a topological space

cutting along a finite number of embedded tori. Each piece is either atoroidal (cannot be cut along an embedded torus in an interesting way) or Seifert-fibered

JSJ decomposition

JSJ_decomposition

3-manifold

Mathematical space

component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered. The acronym JSJ is for William Jaco, Peter Shalen

3-manifold

Geometrization conjecture

Three dimensional analogue of uniformization conjecture

oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition

Geometrization conjecture

Geometrization_conjecture

Virtually fibered conjecture

Conjecture pertaining to finite covers of 3-manifold subfields

mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is

Virtually fibered conjecture

Virtually_fibered_conjecture

Toroidal

Topics referred to by the same term

soil to the atmospheric potential gradient of the "planetary battery". Atoroidal Torus (disambiguation) Sacred Geometry Oligodynamic effect This disambiguation

Toroidal

2π theorem

Gives sufficient condition for Dehn filling to result in a negatively curved 3-manifold

greater than 6 results in a hyperbolike 3-manifold, i.e. an irreducible, atoroidal, non-Seifert-fibered 3-manifold with infinite word hyperbolic fundamental

2π theorem

2π_theorem

Manifold decomposition

Irreducible, orientable, compact 3-manifolds Cut along embedded tori Atoroidal or Seifert-fibered 3-manifolds Union along their boundary, using the trivial

Manifold decomposition

Manifold_decomposition

Orbifold

Generalized manifold

it implies that if X is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, then M has a geometric structure

Orbifold

Fully irreducible automorphism

Concept in mathematics

{\displaystyle \varphi ^{p}} is irreducible. It is known that for any atoroidal φ ∈ Out ⁡ ( F n ) {\displaystyle \varphi \in \operatorname {Out} (F_{n})}

Fully irreducible automorphism

Fully_irreducible_automorphism

Hyperbolic 3-manifold

Manifold of dimension 3 equipped with a hyperbolic metric

compact 3-manifold with toric boundary is irreducible and algebraically atoroidal (meaning that every π1-injectively immersed torus is homotopic to a boundary

Hyperbolic 3-manifold

Hyperbolic_3-manifold

Cannon–Thurston map

an atoroidal fully irreducible element of Out ⁡ ( F n ) {\displaystyle \operatorname {Out} (F_{n})} . Ghosh proved that for an arbitrary atoroidal ϕ ∈

Cannon–Thurston map

Cannon–Thurston_map

Boundary parallel

When a closed manifold embedded in M has an isotopy onto a boundary component of M

{\displaystyle P} is homeomorphic to F × [ 0 , 1 ] {\displaystyle F\times [0,1]} . Atoroidal Satellite knot cf. Definition 3.4.7 in Schultens, Jennifer (2014). Introduction

Boundary parallel

Boundary_parallel

Haken manifold

Mathematics concept

geometrization program for 3-manifolds. Johannson (1979) proved that atoroidal, anannular, boundary-irreducible, Haken three-manifolds have finite mapping

Haken manifold

Haken_manifold