φ For all finite n ∈ N, there is a well-known isomorphism between the standard braid group Bn and the mapping class group π0Hn. This isomorphism has been exhaustively studied in literature, and generalized in many ways. For some basic topological reason, this strong link between finite braid groups and finite mapping class groups can-not be extended to the infinite case in a straightforward way, and, in particular, is not yet well studied in literature. In our work, we define the infinite braid group B∞ to be the group of braids with infinitely many strands, all of which can be possibly nontrivial, i.e., not straight. In particular, this definition does not correspond to the group of finitary infinite braids, which is just the union of all finite braid groups. Similar to the maps π0φn for finite n, we introduce a map that, in particular, turns out not to be an isomorphism. However, we prove its injectivity, and identify its image in B∞. The study of the link between mapping class groups and braid groups in the infinite case is motivated by the study of homeomorphisms in H∞ that give rise to a homoclinic tangle. In fact, the map π0φ∞ attributes to each isotopy class of such a homeomorphism an element of the infinite braid group B∞, and so, allows us to describe the isotopy classes of these homeomorphisms in terms of their image in B∞. Using the fact that the map π0φ∞ is injective, we prove a result that can be applied to the study of the topological structure of homoclinic tangles.
EPFL2008
