By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead given that 1982 whilst Thurston articulated his influential checklist of questions. fundamental between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights comprise the Tameness Theorem of Agol and Calegari-Gabai, the outside Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on particular dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This e-book summarizes a lot of these advancements and gives an exhaustive account of the present state-of-the-art of 3-manifold topology, specially concentrating on the implications for primary teams of 3-manifolds. because the first e-book on 3-manifold topology that includes the fascinating development of the final twenty 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 if a few familiarity with the elemental crew is suggested, little different prior wisdom is thought, and the booklet is obtainable to graduate scholars. The booklet closes with an in depth record of open questions to be able to even be of curiosity to graduate scholars and confirmed researchers. A e-book of the eu Mathematical Society (EMS). allotted in the Americas by way of the yank Mathematical Society.

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On the other hand N has one JSJ-torus, namely, if N is a torus bundle, then the fiber is the JSJ-torus, and if N is a twisted double of K 2 × I, then the JSJ-torus is given by the boundary of K 2 × I. (2) Suppose that N is not a Sol-manifold, and denote by T1 , . . , Tm the JSJ-tori of N. We assume that they are ordered such that the tori T1 , . . , Tn do not bound copies of K 2 × I and that for i = n + 1, . . , m, each Ti cobounds a copy of K 2 × I. Then the geometric decomposition surface of N is given by T1 ∪ · · · ∪ Tn ∪ Kn+1 ∪ · · · ∪ Km .

Then π1 (M1 #M2 ) ∼ = 1 (N1 #N2 ) but in general M1 #M2 is not diffeomorphic to N1 #N2 . Reidemeister [Rer35, p. 109] and Whitehead [Whd41a] classified lens spaces in the PL-category. , the first statement above), then follows from Moise’s proof [Moi52] of the ‘Hauptvermutung’ in dimension 3. Alternatively, this follows from Reidemeister’s argument together with [Chp74]. 1] for more modern accounts and to [Fo52, p. 455], [Bry60, p. 4] for different approaches. The fact that lens spaces with the same fundamental group are not necessarily homeomorphic was first conjectured by Tietze [Tie08, p.

More precisely, if Γ is a 1-manifold in Σ as in the definition of a reducible element in the mapping class group, then the JSJ-tori of N are given by the ϕmapping tori of the 1-manifold Γ, where ϕ is also as in the definition of a reducible element in the mapping class group. In the third case the geometric decomposition of M(Σ) can be obtained by applying the theorem again to the mapping torus of Σ \ νΓ and by iterating this process. 11 3-manifolds with (virtually) solvable fundamental group We finish this chapter by classifying the abelian, nilpotent and solvable groups which appear as fundamental groups of 3-manifolds.

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