By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

ISBN-10: 3037191546

ISBN-13: 9783037191545

The sphere of 3-manifold topology has made nice strides ahead considering the fact that 1982 whilst 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 skin Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on particular dice complexes, and, ultimately, Agol's facts of the digital Haken Conjecture. This e-book summarizes these types of advancements and gives 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 booklet on 3-manifold topology that comes with the intriguing development of the final twenty years, it will likely 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 workforce is usually recommended, little different earlier wisdom is thought, and the booklet is on the market to graduate scholars. The ebook closes with an in depth checklist of open questions that allows you to even be of curiosity to graduate scholars and demonstrated researchers. A e-book of the ecu Mathematical Society (EMS). dispensed in the Americas via the yank Mathematical Society.

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This can be seen by studying the linking form (see [Sei33b, p. 826]) or the Blanchfield form [Bla57], which in turn can be studied using Levine–Tristram signatures (see [Kea73, Lev69, Tri69]). 30 2 Classification of 3-manifolds by their fundamental groups Before we can discuss to what degree the fundamental group determines the homeomorphism of a 3-manifold with boundary we need to introduce a few more definitions. Definition. Let N be a 3-manifold. (1) Suppose N has incompressible boundary. The fundamental group of N together with the set of conjugacy classes of its subgroups determined by the boundary components is called the peripheral structure of N.

If the fundamental group is wordhyperbolic [BaL12, Theorem A]. The high-dimensional results also extend to dimension 4 if the fundamental groups are good in the sense of Freedman [Fre84]. 2; the case where N or N is non-orientable was proved by Heil [Hei69a]. 2 show that fundamental groups determine closed 3-manifolds up to orientation of the prime factors and up to the indeterminacy arising from lens spaces. More precisely, we have the following theorem. 3. Let N and N be closed, oriented 3-manifolds with isomorphic fundamental groups.

Pm , q1 , . . , qm and q1 , . . , qm and oriented manifolds N1 , . . , Nn and N1 , . . , Nn such that (1) we have homeomorphisms N∼ = L(p1 , q1 )# · · · #L(pm , qm ) # N1 # · · · #Nn and ∼ N = L(p1 , q )# · · · #L(pm , q ) # N # · · · #N ; 1 m 1 n (2) Ni and Ni are homeomorphic (but possibly with opposite orientations); and (3) for i = 1, . . , m we have qi ≡ ±q±1 i mod pi . 2, orientable, prime 3-manifolds with infinite fundamental groups are determined by their fundamental groups, provided they are closed.

### 3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

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