By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead considering the fact that 1982 while Thurston articulated his influential record of questions. basic between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights contain the Tameness Theorem of Agol and Calegari-Gabai, the outside Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on specified dice complexes, and, eventually, Agol's evidence of the digital Haken Conjecture. This booklet summarizes some of these advancements and offers an exhaustive account of the present state-of-the-art of 3-manifold topology, in particular concentrating on the results for basic teams of 3-manifolds. because the first e-book on 3-manifold topology that includes the fascinating development of the final twenty years, it will likely be a useful source for researchers within the box who desire a reference for those advancements. It additionally provides a fast paced creation to this fabric. even supposing a few familiarity with the basic workforce is usually recommended, little different past wisdom is thought, and the e-book is offered to graduate scholars. The ebook closes with an in depth checklist of open questions on the way to even be of curiosity to graduate scholars and proven researchers. A booklet of the eu Mathematical Society (EMS). disbursed in the Americas via the yank Mathematical Society.

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We conclude this section with a discussion of 3-manifolds with abelian fundamental groups. In order to do this we first recall the definition of lens spaces. Given coprime natural numbers p and q we denote by L(p, q) the corresponding lens space, defined as L(p, q) := S3 /Z p = {(z1 , z2 ) ∈ C2 | |z1 |2 + |z2 |2 = 1}/Z p , where k ∈ Z p acts on S3 by (z1 , z2 ) → (z1 e2πik/p , z2 e2πikq/p ). Note that π1 (L(p, q)) ∼ = 3 Z p and that the Hopf fibration on S descends to a decomposition of L(p, q) into circles endowing L(p, q) with the structure of a Seifert fibered 3-manifold.

They are diffeomorphic if and only if p1 = p2 and q1 q±1 2 ≡ ±1 mod pi , but they are homotopy equivalent if 2 and only if p1 = p2 and q1 q±1 2 ≡ ±t mod pi for some t, and their fundamental groups are isomorphic if and only if p1 = p2 . 28 2 Classification of 3-manifolds by their fundamental groups (B) Let M, N be compact, oriented 3-manifolds. Denote by N the manifold N with opposite orientation. Then π1 (M#N) ∼ = π1 (M#N) but if neither M nor N has an orientation reversing diffeomorphism, then M#N and M#N are not diffeomorphic.

For 3-manifolds that are the total space of a surface bundle over S1 . Given a compact surface Σ we denote by M(Σ) the mapping class group of Σ, that is, the group of isotopy classes of orientation-preserving self-diffeomorphisms of Σ. For ϕ ∈ M(Σ) we denote by M(Σ, ϕ) = Σ2 × [0, 1] (x, 0) ∼ x, ϕ(x) the corresponding mapping torus with monodromy ϕ. The geometry of a mapping torus can be studied in terms of the monodromy. We start out with the ‘baby case’ that Σ is the torus T 2 = S1 × S1 . 5] for details.