By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead considering 1982 while Thurston articulated his influential record of questions. fundamental between those is Perelman's facts 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 designated dice complexes, and, eventually, Agol's facts of the digital Haken Conjecture. This e-book summarizes these kinds of advancements and gives an exhaustive account of the present state-of-the-art of 3-manifold topology, specially concentrating on the implications for basic teams of 3-manifolds. because the first ebook on 3-manifold topology that comes with the interesting growth of the final 20 years, will probably be a useful source for researchers within the box who desire a reference for those advancements. It additionally supplies a fast paced advent to this fabric. even if a few familiarity with the elemental team is usually recommended, little different earlier wisdom is thought, and the e-book is available to graduate scholars. The booklet closes with an in depth record of open questions with a purpose to even be of curiosity to graduate scholars and verified researchers. A booklet of the ecu Mathematical Society (EMS). dispensed in the Americas via the yank Mathematical Society.
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Additional info for 3-Manifold Groups
It is interesting 32 2 Classification of 3-manifolds by their fundamental groups to ask which topological properties of 3-manifolds can be ‘read off’ from the fundamental group. Given a 3-manifold N we denote by Diff(N) the group of self-diffeomorphisms of N which restrict to the identity on the boundary. Furthermore we denote by Diff0 (N) the identity component of Diff(N). The quotient Diff(N)/Diff0 (N) is denoted by M (N). , the quotient of the group of automorphisms of π by its normal subgroup of inner automorphisms of π).
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.
Somewhat surprisingly, in light of the above discussion, prime knots are in fact determined by their fundamental groups. More precisely, if J1 and J2 are two prime knots with π1 (S3 \ νJ1 ) ∼ = π1 (S3 \ νJ2 ), then there exists a 3 3 homeomorphism f : S → S with f (J1 ) = J2 . 1] (see also [Gra92]) extending earlier work of Culler– Gordon–Luecke–Shalen [CGLS85, CGLS87] and Whitten [Whn86, Whn87]. See [Tie08, Fo52, Neh61a, Sim76b, FeW78, Sim80, Swp80b, Swp86] for earlier discussions and work on this result.