000124757 001__ 124757
000124757 005__ 20181205220240.0
000124757 0247_ $$2doi$$a10.5075/epfl-thesis-4132
000124757 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4132-1
000124757 02471 $$2nebis$$a5573587
000124757 037__ $$aTHESIS
000124757 041__ $$aeng
000124757 088__ $$a4132
000124757 245__ $$aThe link between the infinite mapping class group of the disk and the braid group on infinitely many strands
000124757 269__ $$a2008
000124757 260__ $$aLausanne$$bEPFL$$c2008
000124757 300__ $$a119
000124757 336__ $$aTheses
000124757 520__ $$aφ For all finite n ∈ N, there is a well-known isomorphism  between the standard braid group Bn and the mapping class group π0Hn. This isomorphism has been exhaustively studied in literature, and generalized in many ways. For some basic topological reason, this strong link between finite braid groups and finite mapping class groups can-not be extended to the infinite case in a straightforward way, and, in particular, is not yet well studied in literature. In our work, we define the infinite braid group B∞ to be the group of braids with infinitely many strands, all of which can be possibly nontrivial, i.e., not straight. In particular, this definition does not correspond to the group of finitary infinite braids, which is just the union of all finite braid groups. Similar to the maps π0φn for finite n, we introduce a map  that, in particular, turns out not to be an isomorphism. However, we prove its injectivity, and identify its image in B∞. The study of the link between mapping class groups and braid groups in the infinite case is motivated by the study of homeomorphisms in H∞ that give rise to a homoclinic tangle. In fact, the map π0φ∞ attributes to each isotopy class of such a homeomorphism an element of the infinite braid group B∞, and so, allows us to describe the isotopy classes of these homeomorphisms in terms of their image in B∞. Using the fact that the map π0φ∞ is injective, we prove a result that can be applied to the study of the topological structure of homoclinic tangles.
000124757 6531_ $$ainfinite braid group
000124757 6531_ $$ainfinite mapping class group
000124757 6531_ $$ainfinite permutation group
000124757 6531_ $$ahomoclinic tangles
000124757 6531_ $$agroupe de tresses infini
000124757 6531_ $$amapping class group infini
000124757 6531_ $$agroupe de permutations infinies
000124757 6531_ $$aenchevêtrements homoclines
000124757 700__ $$aBrunner, Jan
000124757 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396
000124757 8564_ $$s1245918$$uhttps://infoscience.epfl.ch/record/124757/files/EPFL_TH4132.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text
000124757 909C0 $$0252139$$pUPHESS$$xU10968
000124757 909CO $$ooai:infoscience.tind.io:124757$$pthesis-bn2018$$pDOI$$pSV$$pthesis$$qDOI2
000124757 919__ $$aGR-HE
000124757 918__ $$aSB$$cIGAT$$dEDMA
000124757 920__ $$b2008
000124757 973__ $$aEPFL$$sPUBLISHED
000124757 970__ $$a4132/THESES
000124757 980__ $$aTHESIS