Atoroidal

In der dreidimensionalen Topologie beschreibt Atoroidalität eine Beziehung zwischen dem Rand einer Mannigfaltigkeit und der Mannigfaltigkeit selbst.

Eine irreduzible Mannigfaltigkeit M heißt geometrisch atoroidal, wenn sich jeder in M inkompressibel eingebettete 2-Torus durch eine Isotopie auf eine Randkomponente von M verschieben lässt. Dies bedeutet, dass M keine eingebetteten Tori enthält, außer solchen, die offensichtlich existieren müssen.

Eine irreduzible Mannigfaltigkeit heißt homotopisch atoroidal, wenn jede Abbildung T^2\to M, die die Fundamentalgruppe des Torus T2 injektiv in die Fundamentalgruppe von M abbildet, zu einer Abbildung in den Rand homotop ist. Dies entspricht der Eigenschaft der Fundamentalgruppe von M, dass jede Untergruppe der Form \mathbb{Z}\times\mathbb{Z} zur Fundamentalgruppe einer Torus-Randkomponente konjugiert ist.

Man kann zeigen, dass "geometrisch atoroidal" aus "homotopisch atoridal" folgt. Die Umkehrung gilt jedoch nicht.

Die Hyperbolisierungvermutung von Thurston besagt, dass jede irreduzible homotopisch atoroidale Mannigfaltigkeit mit unendlicher Fundamentalgruppe eine hyperbolische Struktur trägt.


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