By Christian Kassel, Vladimir Turaev

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

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**Example text**

31 imply that η is injective. Therefore η is an isomorphism. 38. 1 actually shows that the point {idD } is a deformation retract of Top(D). Therefore, πi (Top(D)) = 0 for all i ≥ 0 and the homotopy sequence of the ﬁbration e : Top(D) → Cn (D◦ ) directly implies that the homomorphism ∂ : π1 (Cn , q) → π0 (Top(D, Q)) is an isomorphism. 4 Applications We state two further applications of the techniques introduced above. 39. For any geometric braid b on n strings, the topological type of the pair (R2 × I, b) depends only on n.

3. For n ≥ 1, set Cn = Cn (M ◦ ) = Fn (M ◦ )/Sn . To describe the relation between Top(M ) and Cn , pick a set Q ⊂ M ◦ consisting of n points. We deﬁne an evaluation map e = eQ : Top(M ) → Cn by e(f ) = f (Q), where f ∈ Top(M ). It is easy to deduce from the deﬁnitions that e is a surjective continuous map. 35. The map e : Top(M ) → Cn is a locally trivial ﬁbration with ﬁber Top(M, Q). Proof. Let Fn = Fn (M ◦ ) be the conﬁguration space of n ordered points in M ◦ . We can factor e as the composition of a map c : Top(M ) → Fn with the covering Fn → Cn .

3 The center of Bn The center of a group G is the subgroup of G consisting of all g ∈ G such that gx = xg for every x ∈ G. The center of a group G is denoted by Z(G). 24. If n ≥ 3, then Z(Bn ) = Z(Pn ) is an inﬁnite cyclic group generated by θn = Δ2n , where Δn = (σ1 σ2 · · · σn−1 )(σ1 σ2 · · · σn−2 ) · · · (σ1 σ2 ) σ1 ∈ Bn . 3 Pure braid groups 23 Fig. 11. The braid Δ5 Proof. The braid Δn can be obtained from the trivial braid 1n by a half-twist achieved by keeping the top of the braid ﬁxed and turning over the row of the lower ends by an angle of π.