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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

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

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

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

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

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

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

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

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

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