# Braids and coverings by Vagn Lundsgaard Hansen By Vagn Lundsgaard Hansen

This ebook relies on a graduate direction taught via the writer on the collage of Maryland. The lecture notes were revised and augmented by means of examples. the 1st chapters advance the uncomplicated idea of Artin Braid teams, either geometrically and through homotopy concept, and talk about the hyperlink among knot concept and the combinatorics of braid teams via Markou's Theorem. the ultimate chapters provide a close research of polynomial masking maps, that could be seen as a homomorphism of the elemental workforce of the bottom house into the Artin Braid workforce on n strings.

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We will show that I is a homotopy inverse of g. Let H: (B 1 xB2) xI->-Y H((x 1 ,x 2 ),t) = p(D 1 (x 1 ,t), D2 (x 2 ,t») . 8 Proposition in Switzer  H induces a continuous map H':YxI->-Y such that H' is, of course, base-point preserving. Furthermore, for (x 1 ,x 2 ) EB1xB2: H' (P(x 1 ,x 2 ) ,0) P(x 1 ,x 2 ) H' (P(x 1 ,x 2 ) ,1) p(D 1 (x 1 ,1), D2 (x 2 ,1» It follows that Iog-Id y . Now, since Di(a,t)=a for aEA i , there is a unique continuous map ki:ZixI->-Zi such that ki(qi(x) ,t)=qi (Di(x,t», i=1,2.

H is called a homotopy from f to g. Obviously, - is an equivalence relation on the set of all morphisms from (Y,A) to (Z,B). Moreover, if (W,C) is a third pair and f and g are two morphisms from (Z,B) to (W,C) and both f-g and f-g, then fof-gog. If (Y,A) and (Z,B) are two topological pairs, then (Y,A) is called homotopy equivalent to (Z,B) (we write (Y,A)-(Z,B)) if there are two 27 morphisms f: (Y,A)+(Z,B) and g:(Z,B)+(Y,A) such that fog-1 (Z,B) and gof-1 (Y,A)' where 1 (Y,A) (Y,A) (resp. 1 (Z,B» are the identity morphisms on (resp.

Then G(x)=[D] E u. The proposition is proved. 7 Index and quasi-index pairs. Given an isolating block B, the pair enjoys certain properties which are crucial for the definitions of the Morse and the homotopy indices. These properties are generalized and abstracted in the definitions of index and quasi-index pairs. Intuitively, index and quasi-index pairs are obtained by squeezing or stretching the pair (see Fig. 3). ,," Figure 3 In this section, as usual, X is a metric space and TI is a local semiflow on X.