By Christian Kassel, Vladimir Turaev

During this well-written presentation, influenced by way of various examples and difficulties, the authors introduce the fundamental conception of braid teams, highlighting numerous definitions that exhibit their equivalence; this can be via a remedy of the connection among braids, knots and hyperlinks. very important effects then deal with the linearity and orderability of the topic. suitable extra fabric is integrated in 5 huge appendices. Braid teams will serve graduate scholars and a couple of mathematicians coming from diversified disciplines.

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

**Example text**

To construct c, ﬁx an order in the set Q and deﬁne c by c(f ) = f (Q), where f ∈ Top(M ) and the order in f (Q) is induced by the one in Q. To prove the lemma, it suﬃces to prove that c is a locally trivial ﬁbration. 26. Let us prove the local triviality of c in a neighborhood of a point u0 = (u01 , . . , u0n ) ∈ Fn . For each i = 1, 2, . . , n, pick an open neighborhood Ui ⊂ M ◦ of u0i such that its closure U i is a closed ball with interior Ui . Since the points u01 , . . , u0n are distinct, we can assume that U1 , .

The choice of λ ensures that f u = u on the ball of radius u with center at the origin and f u = 0 outside the ball of radius ( u + 1)/2 with center at the origin. Let {θu,t : U → U }t∈R be the ﬂow determined by f u , that is, the (unique) family of self-diﬀeomorphisms of U such that θu,0 = id and dθu,t (v)/dt = f u (v) for all v ∈ U , t ∈ R. The diﬀeomorphism θu,t smoothly depends on u, t, ﬁxes the sphere ∂U pointwise, and sends the origin to tu. Therefore the map θi : U × U → U deﬁned by θi (u, v) = θu,1 (v) for u ∈ U , v ∈ U satisﬁes all the required conditions.

1 Conﬁguration spaces of ordered sets of points Let M be a topological space and let Mn = M × M × · · · × M be the product of n ≥ 1 copies of M with the product topology. Set Fn (M ) = {(u1 , u2 , . . , un ) ∈ M n | ui = uj for all i = j} . This subspace of M n is called the conﬁguration space of ordered n-tuples of (distinct) points in M . If M is a topological manifold (possibly with boundary ∂M ), then the conﬁguration space Fn (M ) is a topological manifold of dimension n dim(M ). Clearly, any ordered n-tuple of points in M can be deformed into an ordered n-tuple of points in the interior M ◦ = M − ∂M of M .