By Christian Kassel, Vladimir Turaev

ISBN-10: 0387338411

ISBN-13: 9780387338415

During this well-written presentation, encouraged through quite a few examples and difficulties, the authors introduce the elemental idea of braid teams, highlighting numerous definitions that express their equivalence; this can be through a remedy of the connection among braids, knots and hyperlinks. vital effects then deal with the linearity and orderability of the topic. proper extra fabric is incorporated in 5 huge appendices. Braid teams will serve graduate scholars and a couple of mathematicians coming from different disciplines.

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**Extra resources for Braid Groups (Graduate Texts in Mathematics, Volume 247)**

**Example text**

For detailed expositions of knot theory, the reader is referred to the monographs [BZ85], [Kaw96], [Mur96], [Rol76]. 1 Basic deﬁnitions Let M be a 3-dimensional topological manifold, possibly with boundary ∂M . A geometric link in M is a locally ﬂat closed 1-dimensional submanifold of M . Recall that a manifold is closed if it is compact and has an empty boundary. A closed 1-dimensional submanifold L ⊂ M is locally ﬂat if every point of L has a neighborhood U ⊂ M such that the pair (U, U ∩ L) is homeomorphic to the pair (R3 , R × {0} × {0}).

39. For any geometric braid b on n strings, the topological type of the pair (R2 × I, b) depends only on n. Proof. Pick a disk D ⊂ R2 such that b ⊂ D◦ × I. 16) lies in D◦ . 37, there is a normal isotopy {ft : D → D}t∈I parametrizing b. The formula (x, t) → (ft (x), t) deﬁnes a homeomorphism F : D × I → D × I mapping Q × I onto b and keeping ∂D × I pointwise. Extending F by the identity on (R2 − D) × I, we obtain a homeomorphism R2 × I → R2 × I mapping Q × I onto b. Note that this homeomorphism is level-preserving in the sense that it commutes with the projection to I.

Let M be a connected topological manifold of dimension ≥ 2 with ∂M = ∅. For any m ≥ 0, n > r ≥ 1, the forgetting map p : Fm,n (M ) → Fm,r (M ) deﬁned by p(u1 , . . , un ) = (u1 , . . , ur ) is a locally trivial ﬁbration with ﬁber Fm+r,n−r (M ). Proof. 26 to M − Qm . 28 1 Braids and Braid Groups Recall that a connected manifold M is aspherical if its universal covering is contractible or, equivalently, if its homotopy groups πi (M ) vanish for all i ≥ 2. 28. For any m ≥ 0, n ≥ 1, the manifold Fm,n (R2 ) is aspherical.