Una presentazione dei gruppi delle trecce

The braid groups can be thought of as a generalisation of the symmetric group, in which not only the permutation but also the direction of exchange of points is considered; the consequence of this is that transpositions no longer have order two, indeed in the braid groups there are no elements of finite order. In this paper we follow the idea of Fox R. and Neuwirth L. to discuss a proof of the presentation of the braid groups. First of all we recall some prerequisites: mainly results on groups, CW complexes, foundamental groups and covering spaces. Then we give the proof of this presentation, obtained by considering the braid groups as foundamental groups of certain configuration spaces. Finally, we have also derived a presentation of the symmetric group, exploiting a short exact sequence involving pure braids, braids and permutations. We belive that the main contribution of this dissertation could be the wealth of examples, proofs and references which, in our opinion, were scarce in the original paper, to the point of making necessary the modification of some otherwise imprecise constructions.