# Braids and coverings by Vagn Lundsgaard Hansen

By Vagn Lundsgaard Hansen

This e-book is predicated on a graduate path taught by way of the writer on the college of Maryland. The lecture notes were revised and augmented via examples. the 1st chapters boost the user-friendly conception of Artin Braid teams, either geometrically and through homotopy idea, and speak about the hyperlink among knot conception and the combinatorics of braid teams via Markou's Theorem. the ultimate chapters supply an in depth research of polynomial masking maps, that may be considered as a homomorphism of the basic staff of the bottom house into the Artin Braid team on n strings.

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C / Â K ! Fn /: The generalized Brouwer theorem tells us that fn has fixed points; choose one of them and call it yn . yn /g to avoid messy subscripts. Call the limit of the subsequence y and note that it is in the closed set C . We claim that y is a fixed point of f . x/; x// < n1 . yn /g does, namely y. y/ D y. u t A map on a compact domain is of course compact, so the Schauder theorem implies the Brouwer fixed point theorem. It also implies the following generalization of the Brouwer theorem which is the fixed point theorem that we used in the topological proof in Chap.

We will require as part of the setting of the Brouwer degree that F is an admissible subset of U . Think of S n as Rn [ 1 so that U is a subset of S n . S n / ! U; U F/ F /. The excision property of homology F / ! S n ; S n F/ induces an isomorphism of homology. U; U F / by setting j 1 k . S n / is that generator that we just chose so carefully. U; U F /, but we can be sure that 0n is nontrivial, provided only that F is nonempty, for the following reason. S n i k F / ! S n / ! S n F / ! S n fxg/ !

C Œ0; 1 is the second derivative operator. 4 we know that the function S defined as the composition j F L 1 S W C02 Œ0; 1 ! C 1 Œ0; 1 ! C Œ0; 1 ! C02 Œ0; 1 is completely continuous. We’ll still prove that the boundary value problem has a solution by proving that S must have a fixed point. But the argument has to be different than the one we used in Chap. 5. F. s; u; p/ D u C p 2 satisfies the conditions from Chap. 6 with, say, M D 1, but its image is all of R. 6) that we obtained from the Schauder theorem will assure us that S has a fixed point.