By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead seeing that 1982 whilst Thurston articulated his influential checklist of questions. basic between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights comprise the Tameness Theorem of Agol and Calegari-Gabai, the skin 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 e-book summarizes these kinds of advancements and gives an exhaustive account of the present cutting-edge of 3-manifold topology, specially concentrating on the implications for primary teams of 3-manifolds. because the first booklet on 3-manifold topology that comes with the interesting growth 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 provides a fast moving creation to this fabric. even supposing a few familiarity with the elemental workforce is usually recommended, little different prior wisdom is believed, and the e-book is obtainable to graduate scholars. The ebook closes with an in depth record of open questions with a purpose to even be of curiosity to graduate scholars and verified researchers. A book of the eu Mathematical Society (EMS). disbursed in the Americas through the yankee Mathematical Society.
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Extra resources for 3-Manifold Groups
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. 117], [Vo13a] and was first proved by Alexander [Ale19, Ale24a]; see also [Sti93, p.
2, orientable, prime 3-manifolds with infinite fundamental groups are determined by their fundamental groups, provided they are closed. The same conclusion does not hold if we allow boundary. , the exteriors of the granny knot and the square knot) have isomorphic fundamental groups, but the spaces are not homeomorphic. 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]).
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. (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 , .
3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton