By Jung Rae Cho, Jens Mennicke

ISBN-10: 388538227X

ISBN-13: 9783885382270

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Xi_I)Xi,(Xi+t}xp· ··,(xn)x;) for 1 ~ i ~ n. For details here see [FRS]. s(X) , the associated group of X . Proof Since SeX) has only one vertex the fundamental group of the 1skeleton is the free group F(X). Adding the 2-cells is equivalent to adding 0 the relations ab = b-Iab which gives the required result. We now investigate the homotopy group 11"2 (EX). To do this we first find a combinatorial representation of 11"21 N (S) I for a general species S. Many of the concepts such as labelled diagrams can be generalised to give descriptions of the higher homotopy groups 1I"nIN(S)1 where n > 2 and of the bordism groups nnIN(S)I.

Vertical Composition: Preferred squares a, b, c, d and a', b', c', (1' with c = a' may be composed vertically to form the preferred square a, bb', c', dd' with base a. b8 c a c' c' d' a' b' 0 d' b 0 d a Composing Squares S5. re unique c, d, D so that C C --t D fb o a jd A --t B is preferred. re determined by two binary operations c = a b and d = ba • b~ a 43 V2 In the following diagram, if the three squares centre, south and west are given and preferred then the diagram can be completed as shown so that the squares; outside, east and north are preferred.

3 21 identify the 2-faces of the pyramid by non-Euclidean motions in such a way that equally arrowed edges coincide. It can be shown directly that the result is a 3-manifold homeomorphic to the complement S3 - 41 . It inherits a hyperbolic structure from H3 which is the universal covering of S3 - 41 , The group of the four knot thus turns out to be a discrete subgroup of PSL(2, <11, the group of isometries of H3, with a fundamental domain of finite volume. By the so-called rigidity theorem of Mostow the hyperbolic structure is - up to isometries - uniquely determined by the link type.

### Recent Advances in Group Theory and Low-Dimensional Topology by Jung Rae Cho, Jens Mennicke

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