By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead on account that 1982 whilst Thurston articulated his influential record of questions. basic between those is Perelman's facts of the Geometrization Conjecture, yet different highlights contain the Tameness Theorem of Agol and Calegari-Gabai, the skin Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on exact dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This ebook summarizes these kinds of advancements and gives an exhaustive account of the present state-of-the-art of 3-manifold topology, particularly concentrating on the implications for primary teams of 3-manifolds. because the first publication on 3-manifold topology that comes with the interesting development of the final 20 years, will probably be a useful source for researchers within the box who want a reference for those advancements. It additionally offers a fast paced creation to this fabric. even though a few familiarity with the elemental staff is suggested, little different past wisdom is thought, and the booklet is obtainable to graduate scholars. The e-book closes with an in depth record of open questions with the intention to even be of curiosity to graduate scholars and validated researchers. A ebook of the ecu Mathematical Society (EMS). dispensed in the Americas through the yankee Mathematical Society.
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For example, if N is a Sol-manifold, then N is also a Sol-manifold. A Sol-manifold has precisely one JSJ-torus, but the preimage of the JSJ-torus of N under the map p might have many components. 3. Proof. The fact that N is again irreducible follows from the Equivariant Sphere Theorem (see [MSY82, p. 647] and also [Duw85, Ed86, JR89]). ) If N is geometric, then N is clearly also geometric, and there is nothing to prove. We henceforth assume that N is not geometric. 2 that N is not a twisted double of K 2 × I.
The Kneser Conjecture implies that the converse holds. 1 (Kneser Conjecture). Let N be a compact, oriented 3-manifold with incompressible boundary. If π1 (N) ∼ = Γ1 ∗ Γ2 , then there exist compact, oriented 3-manifolds N1 and N2 with π1 (Ni ) ∼ = Γi for i = 1, 2 and N ∼ = N1 #N2 . This theorem was first stated by Kneser [Kn29], with a (rather obscure) proof relying on Dehn’s Lemma. The statement was named ‘Kneser’s Conjecture’ by Papakyriakopoulos. The Kneser Conjecture was first proved completely by Stallings [Sta59a, Sta59b] in the closed case, and by Heil [Hei72, p.
2]), Boileau–Otal [BO86], [BO91, Th´eor`eme 3], and the Geometrization Theorem. (1) A Seifert fibered manifold is called small if it is not Haken. 15]. Inspection of trivial cases and [McC91, p. 21] show that if N is a small Seifert manifold, then Out(π1 (N)) is finite. On the other hand, if N is Seifert fibered but not small, then the map Φ above is in fact an isomorphism. (2) Let N be a hyperbolic 3-manifold. As a consequence of the Rigidity Theorem, Out(π1 (N)) is finite and naturally isomorphic to the isometry group of N.