By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead given that 1982 while Thurston articulated his influential checklist of questions. basic between those is Perelman's facts 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 detailed dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This booklet summarizes these kinds of advancements and gives an exhaustive account of the present cutting-edge of 3-manifold topology, in particular concentrating on the implications for primary teams of 3-manifolds. because the first publication on 3-manifold topology that includes the intriguing development of the final 20 years, it is going to be a useful source for researchers within the box who want a reference for those advancements. It additionally supplies a fast moving creation to this fabric. even though a few familiarity with the basic staff is usually recommended, little different earlier wisdom is believed, and the booklet is obtainable to graduate scholars. The booklet closes with an in depth checklist of open questions as a way to even be of curiosity to graduate scholars and proven researchers. A book of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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**Sample text**

1 says that if #T = #T , then T and T are isotopic. It follows easily that S and S are also isotopic. 1 as the geometric decomposition surface of N. 6 and vice versa. 1 gives us the following proposition. 2. Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary with N = S1 × D2 , N = T 2 × I, and N = K 2 × I. , the geometric decomposition surface is empty. 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.

By the Elliptization Theorem, any 3-manifold M with π1 (M) ∼ = Z2 is diffeomorphic to RP3 . It follows that if N has solvable fundamental group, then either N ∼ = RP3 #RP3 or N is prime. 1 2 Since S ×S is the only orientable prime 3-manifold which is not irreducible we can henceforth assume that N is irreducible. Now let N be an irreducible 3-manifold such that no boundary component is a 2-sphere and such that π = π1 (N) is infinite and solvable. 2 and the Poincar´e Conjecture show that our assumption that π1 (N) is virtually solvable implies that one of the following occurs: either N = S1 × D2 or the boundary of N is incompressible.

There is a (possibly empty) collection of disjointly embedded incompressible surfaces S1 , . . , Sm which are either tori or Klein bottles, such that each component of N cut along S1 ∪ · · · ∪ Sm is geometric. Any such collection with a minimal number of components is unique up to isotopy. Proof. If N is already geometric, then there is nothing to prove. For the remainder of the proof we therefore assume that N is not geometric. 1 we can thus in particular assume that N is not a torus bundle. 2 that N is not a twisted double of K 2 × I.